Control-Moment Gyroscopes for Joint Actuation: A New Paradigm …€¦ · Princeton Satellite Systems, Princeton, New Jersey, 08540, USA Manned spacecraft will require maintenance

Control-Moment Gyroscopes for Joint Actuation: A

New Paradigm in Space Robotics

Mason A. Peck∗

Cornell University, Ithaca, New York, 14853, USA

Michael A. Paluszek†, Stephanie J. Thomas‡, and Joseph B. Mueller§

Princeton Satellite Systems, Princeton, New Jersey, 08540, USA

Manned spacecraft will require maintenance robots to inspect and repair componentsof the spacecraft that are only accessible from the outside. This paper presents a designof a novel free-flying maintenance robot (known as a MaintenanceBot.) The Maintenance-Bot uses Control Moment Gyros (CMGs) for manipulator arm and attitude control. Thisprovides high authority control in a compact low power package. Relative position and at-titude determination is accomplished with an RF system supplemented by a vision systemat close range. When not docked to the manned vehicle (which must be done periodicallyto refuel and recharge batteries or when the manned vehicle performs orbit changes) theMaintenanceBots fly in formation using a cold gas thruster system and formation flyingalgorithms that permit dozens of MaintenanceBots to coordinate their positions. The useof CMGs is a prominent feature of this design. An array of CMGs can exchange angularmomentum with the spacecraft body to effect attitude changes, as long as certain math-ematical singularities in the actuator Jacobian are avoided. The proposed maintenancerobot benefits dramatically from the dynamics and control of a multibody robotic armwhose joints are driven by CMGs. In addition to high power efficiency, another advantageof this concept is that spacecraft appendages actuated by CMGs can be considered reac-tionless, in the sense that careful manipulation of the CMG gimbal angles can virtuallyeliminate moments applied to the spacecraft body. This paper provides a preliminary de-sign of the MaintenanceBot. Analysis of the formation flying and close maneuver controlsystems is included. Simulation results for a typical operation is provided.

I. Introduction

Free flying maintenance robots have the potential to greatly increase the safety of future manned missions.This paper discusses a concept for an agile, free-flying maintenance, assembly, and servicing robot based

on the formation and close-maneuver software developed at Princeton Satellite Systems, visualization systemsdeveloped at PSS and control-moment gyro (CMG) attitude and effector control in development at CornellUniversity. Because of their high agility and low power requirements, these robots will help improve systemreliability by allowing in-situ repairs for which an astronaut would otherwise require extensive training andwill be able to undertake tasks that would otherwise be too dangerous or require too much mechanicalstrength for an astronaut.

Many robotic manipulators have been proposed for space applications. Robonaut1 developed at NASA’sJohnson Space Center in Houston, Texas, has five-fingered, multi-jointed hands. Ranger, developed at theUniversity of Maryland in College Park has tools that plug into each wrist socket. Dextre,2 a product ofMD Robotics in Brampton, Ontario, will be launched to the International Space Station in 2007. Dextre isa complex robot designed to perform intricate maintenance and servicing tasks on the outside of the ISS.Dextre works by grabbing an ISS stabilization point to anchor itself. This is in contrast to MaintenanceBot’s

∗Professor, Mechanical and Aerospace Engineering, 212 Upson Hall, Cornell University, and Senior Member AIAA.†President, Princeton Satellite Systems, 33 Witherspoon Street, Princeton NJ, 08540, USA Senior Member AIAA.‡Senior Technical Staff, Princeton Satellite Systems, 33 Witherspoon Street, Princeton NJ, 08540, USA Member AIAA.§Senior Technical Staff, Princeton Satellite Systems, 33 Witherspoon Street, Princeton NJ, 08540, USA Member AIAA.

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American Institute of Aeronautics and Astronautics

high torque CMGs which reduce the need for anchoring allowing it to operate around any vehicle and atany location around the vehicle. Dextre is shown in Figure 1. Momentum unloading of the Single GimbalControl Moment Gyros (SCMGs) is still most effectively accomplished by anchoring the MaintenanceBot tothe spacecraft and unloading the SCMGs.

Figure 1. Dextre

The MaintenanceBots directly address several of the challenges thatneed to be met for sustained manned exploration:

• They represent a radically low-power and high-redundancy meansof in-space assembly and maintenance

• Their sophisticated software and their low power ensures long-duration autonomous operation

• They are designed for reusability, to follow a larger spacecraft andperform servicing tasks on demand

• They can be reconfigured for specific assembly tasks, merely byselecting different end effectors; furthermore, the scalability ofthis technology allows the MaintenanceBot software and mission-operations procedures to be reused in a modular fashion for multipletasks

• Multiple CMGs for attitude and manipulator actuation ensuressystem-wide mechanical and control redundancy; the network-centric object-oriented flight-operation software ensures margin andredundancy at the operational level.

Each of these MaintenanceBots features low power, but high-torque, actuation not only of its attitude,but also of its manipulators, which are also driven with CMGs. CMGs are a high-TRL technology that canoffer up to 5 Nm per watt of electrical power. The novel, multiple-redundant use of CMGs for attitude anddexterous control distinguishes this concept from classical on-axis actuation of robotic arms, which leadsto low torque per mass and low torque per Watt. High-agility servicing systems will be key to providingmultipurpose, quick turn-around capabilities across the many on-orbit platforms that motivate this BAA.The dynamics and control of the CMG-actuated MaintenanceBot are readily scalable, and the scaled systemretains its promise of high torque for low power. Maturing this system will involve integrating these mid-TRLtechnologies (CMGs and agile formation flight software) into a flight-qualified system and demonstrating theiroperation in the micro gravity environment offered on the shuttle or ISS.

The close-maneuver algorithms are based on the A* optimal path-planning algorithm which provides theflexibility needed for maneuvering around a large, irregularly shaped vehicle such as a lunar-transfer vehicleor space station. Formation flying algorithms are based on PSS work with NASA/GSFC and AFRL.

II. Requirements

No formal requirements exist for a maintenance robot. For the purposes of this paper, a set of level 1requirements were developed. These are presented in Table 1 on the following page. These provide a basisfor the design of the MaintenanceBot in this paper.

Requirement 1 is the minimum functionality required. This alone is a necessary function that would haveproven invaluable on several manned space missions. Requirement 2 is the next step up in functionality. Atthis stage we assume that the components are designed for robotic or astronaut replacement. Requirements3-7 define the operational capabilities of the MaintenanceBot. Requirement 7 permits the simultaneousoperation of many MaintenanceBots. This requirement implies that the MaintenanceBots, when not undercrew member control, will not get in each other’s way.

Requirement 8 provides the basis for the propellant budget of the system. It also provides the requirementfor the sizing of the thrusters which need to produce a radial force to maintain the circular orbit. Thisrequirement should be considered a starting point and will need to be revised when the operational detailsof the manned spacecraft are better developed.

Requirement 9 is quite general and expresses an ideal. The design has the crew controlling the manipulatorarm operations. It is desirable to minimize the degree of interaction required. For example, it is preferable

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Table 1. MaintenanceBot requirements

Level 1 Requirement

1.0 Provide autonomous visual surveillance of the exterior of the manned spacecraft

2.0 Remove and replace external components under astronaut control

3.0 Operate autonomously when the manned spacecraft is coasting

4.0 Recharge batteries and refuel from the manned spacecraft

5.0 Autonomously dock with the manned spacecraft

6.0 Autonomously maneuver in close proximity to the manned spacecraft

7.0 Maintain formation with any number of MaintenanceBots

8.0 Orbit manned spacecraft once per hour at a radius of 50 m for up to 1 day

9.0 Provide a seamless link between autonomous and crew operation of the MaintenanceBot

10.0 Be repairable by the crew

to have the crew command to grasp the bolt rather than actually move the arm using joysticks or by meansof sensors attached to the crew members arm.

Requirement 10 requires that it be possible to bring the MaintenanceBot onboard the manned spacecraftand repair it by replacing components, fixing the wiring, etc. This rules out the use of any dangerousfuels, such as hydrazine and that it be small enough to fit inside the habitable volume while repair work isunderway.

III. MaintenanceBot Architecture

The architecture is shown in Figure 2 including major hardware and software components.Relative sensing is performed with the camera, intersatellite links and a radio frequency location system.

The latter provide range and range rate while the camera provides relative attitude and position information.Camera processing is lumped in the camera block. The outputs of the ISL, camera and RF system blocksis used to determine the relative state and attitude. Relative state is with respect to all spacecraft in thevicinity while relative attitude is with respect to the manned spacecraft. The crew interface is also carriedvia the ISL.

The maneuver planner can command formation flying or close maneuvers depending on the particulartasks selected by the crew. The close maneuver and formation flying blocks drive the orbit control blocks.Effectively they compute orbit errors which must be nulled by orbit control. Close maneuver also includesdocking and separation. The crew also drives the arm control and the attitude control system. Althoughshown as separate blocks since attitude control can be performed with the arm fixed, the system is actuallyperforming n-dof control of both the arm and core.

The momentum control block operates in background and commands momentum unloading as needed.This can be accomplished through thruster firing or through grabbing hold of the manned spacecraft withone of the arms.

The momentum, arm and attitude control drive the SCMGs through a torque distribution algorithm. Aforce and torque distribution algorithm uses simplex (linear programming) to distribute 6 dof commands to 6+ m thrusters where m is any number 1 or larger. This latter algorithm was used onboard the Cakrawarta-1spacecraft.? Simplex solves the equation

Au = b, u ≥ 0 (1)

subject to the constraint to minimize cu where there are more columns of A then rows of b.

IV. Low-Power, Reactionless Attitude and Joint Control with CMGs

A. Theoretical Development

Control-moment gyroscopes (CMGs) have been used for attitude control on spacecraft that require largetorques. A CMG consists of a spinning rotor and one or more motorized gimbals that tilt the rotors angularmomentum. As the rotor tilts, the changing angular momentum causes a gyroscopic torque that rotates the

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M a i n t e n c e B o t sM a n n e d S C I S LC a m e r a M a n e u v e rP l a n n e rC l o s e M a n e u v e r F o r m a t i o n F l y i n g

S t a t eD e t e r m i n a t i o n

O r b i t C o n t r o lA t t i t u d e C o n t r o lC r e w I n t e r f a c e

T o r q u eD i s t r i b u t i o nA r m C o n t r o l

S C M G s F o r c e D s t r i b u t i o nT h r u s t e r s

M o m e n t u mC o n t r o lG y r o sO t h e r S p a c e c r a f t

Figure 2. MaintenanceBot architecture

spacecraft. Figure 3 is a picture of the Honeywell M50 CMG, which produces 50 ft-lb with 50 ft-lb-sec rotoron a 1 rad/sec gimbal. Achieving this peak torque requires an astonishingly low 120W.3

CMGs have been used for decades in large spacecraft, including Skylab and the International SpaceStation. These large spacecraft have implemented dual-gimbal CMGs, which do not offer the power, torque,and robustness benefits of the single-gimbal CMGs envisioned for the MaintenanceBot. In the years to comesingle-gimbal CMGs will provide attitude control for several commercial earth-imaging satellites, such asLockheed-Martin’s Ikonos and Ball Aerospace’s WorldView spacecraft. We propose to base the design ofthe CMGs for the MaintenanceBot on the technology of Honeywell Defense and Space Electronics Systems,whose high-reliability and high-TRL CMGs boast an unrivaled history of mission success.

A CMG is far more power efficient than the conceptually simpler and more commonly used reactionwheel (RWA). An RWA applies torque simply by changing its rotor spin speed ωs, but in doing so impartsshaft power Ps

Ps = τ · ωs (2)

a result that assumes an entirely mechanical, lossless system. The scalar Ps is the projection of the vectormotor torque τ onto the rotor angular-velocity vector ω. In contrast, a CMGs gimbal motor is roughlyorthogonal to the rotor spin axis, and the resulting shaft power is virtually zero if the gimbal inertia and themotor losses are negligible. For a few hundred watts and about 100 kg of mass, large CMGs have producedthousands of Nm of torque, enough to flip over an SUV.4 A reaction wheel of similar capability would requiremegawatts of power to produce torque at speed.

Operating either a CMG or an RWA produces a torque that reacts onto the spacecraft body, influencingthe spacecraft angular momentum. The difference is that the CMG’s own angular momentum changes in

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Figure 3. Honeywell M50 CMG

direction (but not in magnitude), while the RWA’s angular momentum changes in magnitude (but not indirection). If the CMG’s gimbal is rigid, the gyroscopic torque for which a CMG is responsible is purely aconstraint torque. As such, it does no work. At the heart of this surprising result is the counterintuitive factthat one can alter the distribution of momentum among bodies in a dynamical system in a way that is inde-pendent of energy. That is, changing the angular momentum of various links in the MaintenanceBot systemcan be done in a way that requires no energy except what is lost through electromechanical inefficiencies.Using CMGs is a natural way to realize such an architecture.

Momentum-storage devices (to date, exclusively reaction wheels) have been used to provide reactionlessmotion of high-agility gimbals and entire spacecraft payloads. The principle of operation is simple: ratherthan using a motor that reacts the drive torque of a moving component back onto the spacecraft bus, whereit must be dealt with as an attitude disturbance, a reactionless drive absorbs the momentum internally. Forexample, a gimbal may be actuated by an RWA (realized, perhaps, as a circumferential ring) aligned withthe gimbal axis. When the RWA spins up in one direction, the gimbal spins up in the other. The concept isshown in Figure 4.

Figure 4. Schematic of a Reactionless Jointed System

One can generalize this single-axis principle to multiple degrees of freedom associated with a spacecraftpayload: a collection of RWAs manages the entire payloads angular momentum state in three degrees offreedom, so that the payload may undergo attitude motions that are largely imperceptible to the rest ofthe spacecraft. Such an architecture simplifies design and integration because payload components maybe developed independently, without the risk of unwanted interactions after the system is built. It alsosimplifies operations. Many tasks can be undertaken simultaneously with virtually no coupling betweenphysical behaviors or tasking. In the case of the MaintenanceBot, these tasks may include simultaneouslymanipulating many different components of a spacecraft under repair.

Reactionless benefits come with the CMG-driven MaintenanceBot. Here, incorporating CMGs through-out the kinematic chain of links provides not only high torque actuation but also inherent reactionlessdynamics. When a joint is actuated, the torque comes not from a direct-drive motor that interacts with itsneighboring body but from manipulating the distribution of momentum among the CMGs and the body towhich they are mounted. Perhaps the most important impact of reactionless CMG-based control is that itrequires only 1%-10% the electrical power for a comparable RWA-based or joint-driven robotic system, as

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we explain in this section. This feature enables high agility (or dexterity) for modest power or typical agilityfor considerable power savings over existing robotic concepts. Low power improves the robustness and safetyof the MaintenanceBot system, and the resulting cost and mass savings likely roll up to the system level.

Although many CMG arrangements are possible, our baseline concept includes a scissored pair of CMGsfor each rotational joint in the MaintenanceBot. A scissored pair is an array of two CMGs with parallelgimbal axes and opposite angular velocities. Equivalently, a scissored pair may be said to consist of twoCMGs with antiparallel gimbal axes and equal angular velocities. Figure 5 is a sketch of the concept. Thescissored-pair arrangement ensures that the sum of the CMGs angular momentum aligns with a single axis,like that of a reaction wheel, which drastically simplifies the control algorithms. As Figure 5 indicates, therelative angle between the CMG gimbal angles φ1 and φ2 are kept constant, either through mechanical meanssuch as gears or through closed-loop control. Although the individual CMGs angular-momentum vectors hr1

and hr2 tilt away from the rotational joints axis a, their sum remains aligned with a. Thus, the momentumexchange from the CMG to the jointed body accelerates the body about its joint axis only, without couplinginto the rest of the base body A.

Figure 5. Scissored Pair of CMGs and Implementation on a Single-DOF Jointed Body

The robotic arm(s) of the MaintenanceBot may consist of many such jointed bodies. A schematic isshown in Figure 6, where the end-effector is represented as a black cone (suggesting, perhaps, the sun shieldfor the lens of an inspection camera).

Figure 6. Three-body Robotic Arm, Each Joint Actuated by a Scissored Pair of CMGs

If necessary the base body can be driven by a suite of four identical CMGs. This choice allows a singlemathematical degree of freedom for singularity avoidance, providing a singularity-free region of operationwithin a 2hb sphere, where hb is the momentum of an individual CMG in the base body.

The low-power benefits of CMGs are most effectively realized in a system where the CMG base’s angular-velocity vector does not cause gyroscopic torques along the CMG gimbal axes, which would introduce largeholding-torque requirements on the gimbal motors. Conversely, reactionless control of outboard joints vir-tually eliminates control power required on inboard joints (and the base body). For example, consider a

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Figure 7. Two-Link Example with Two Scissored Pairs of CMGs

situation in which a joint rotates about an inertial axis that is perpendicular to the CMG momentum vectorand to the CMG gimbal axis. In the case of the single body in Figure 6, one such axis is the joint axis whenthe CMG has rotated about its gimbal roughly 90 deg from the orientation shown. Such rotation appliestorque back onto the CMG gimbal motor, which the motor then must react. Unless the motor has somemechanical antibacklash device (like a ratchet), the motor must apply this torque through electrical powerin its windings with all the related losses and impacts to harness design.

B. CMG Failures

The MaintenanceBot architecture can accommodate a variety of approaches to subsystem fault tolerance.With regard to CMG failures or underperformance, we propose a concept in which the base body attitudecontrol is coupled to the joint attitude control to minimize power (or keep power within constraints fordegraded agility). Again, the power-optimal design is one that includes reactionless control of the joints.However, in a multibody system without reactionless control, it is possible to describe constraints on themotion such that a steering algorithm can minimize power. A detailed explanation follows.

We consider the case of n jointed bodies, each of which is driven by a single CMG. The scissored-pairconfiguration is subject to the same constraints, but the example is clearer for single CMGs, and thereforewe proceed with the simpler case. Let ωi/N represent the ith bodys angular-velocity vector in an inertialframe N . Let gi represent the ith CMGs gimbal axis, and let hri represent the ith CMGs angular-momentumvector. The angle φi is the ith CMGs gimbal axis, taken to be zero when hi ‖ ai , the ith body’s joint axis.We define a frame B that rotates with the base body (the central body of the maintenance bot). The totalangular velocity of the ith body is then the sum of the joint rates along the kinematic chain back to the basebody, whose inertial angular velocity is ωB/N

ωB/N = ωi/i−1 + ωi−1/i−2 + ... + ω2/1 + ω1/B + ωB/N (3)

We assume that the CMG gimbal axes are perpendicular to the joint axes (i.e. gi ⊥ ai). Expressedalgebraically, the requirement for minimum power is then

(

ωi/n × hri

)

· gi = 0 (4)

i.e., the projection of the gyroscopic torque due to the inertial rate of the joint onto the CMG gimbal axis iszero. Writing these constraints for all of the joints in the system results in n equations in the n joint angular

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rates θ

‖hrn‖ sinφn gn · (hrn × an−1) gn · (hrn × an−2) . . . gn · (hrn × a1)

‖hrn−1‖ sinφn−1 0 gn−1 · (hrn−1 × an−2) . . . gn−1 · (hrn−1 × a1)...

.... . .

. . ....

‖hr1‖ sinφ1 0 0 0 g2 · (hr2 × a1)

θn

θn−1

...

θ1

=

−gn ·(

hrn × ωB/N)

−gn−1 ·(

hrn−1 × ωB/N)

...

−g1 ·(

hr1 × ωB/N)

(5)This expression is written without explicit basis vectors. Implementation would likely include a coordinate

system based on Denavit-Hartenberg parameters or some other convenient rubric for coordinatizing theforward kinematics. We also note here that if the rotor angular-momentum is the same from one CMG tothe next, its magnitude can be factored out of this equation, resulting in a purely kinematical expression.

Solving these equations for θ involves inverting an n×n matrix because these constraints specify each jointrate. However, it is clear that this matrix can be singular for certain joint alignments. As a demonstration,we represent them in the form of an (n − 1) × (n − 1) system of equations and a single scalar equation:

gn · (hrn × an−1) gn · (hrn × an−2) . . . gn · (hrn × a1)

0 gn−1 · (hrn−1 × an−2) . . . gn−1 · (hrn−1 × a1)...

. . .. . .

...

0 0 . . . g2 · (hr2 × a1)

θn

θn−1

...

θ1

=

‖hrn‖θn sin φn − gn ·(

hrn × ωB/N)

‖hrn−1‖θn−1 sin φn−1 − gn−1 ·(

hrn−1 × ωB/N)

...

‖hr1‖θ1 sinφ1 − g1 ·(

hr1 × ωB/N)

(6)

whereg1 ·

(

hr1 × ωB/N)

= ‖hr1‖θ1 sinφ1 (7)

Consider the case of sinφ1 = 0 Here, the inboardmost joint rate θ1 is undefined unless g1 ⊥(

hr1 × ωB/N)

or hr1 ‖ ωB/N , in which case θ1 is merely unconstrained and can therefore take on any value. This principlecan be extended for all of the joint angles, although simple expressions are not available (they involve thesingular values of the matrix).

The fifth equation constrains the central-body rate such that the inboardmost joint requires zero power.The presence of this constraint forces the attitude-control systems engineer to consider an important question:given multiple kinematic chains (robotic arms), each of which may be attached to the central body, whatis the best strategy for accommodating the minimum-power requirements of all of them simultaneously?One approach is to constrain either the joint rate θ1 or the gimbal angle φ1 of each arm so that the fifthequation is satisfied. However, doing so essentially prevents the innermost joint from moving to place theend effector in a desired trajectory; it also introduces a constraint that cascades throughout the rest of thebodies, constraining their motion as well. The solution must weigh the power-minimization needs of alljoints and the central body simultaneously with the combined attitude-control and joint-control commandsto be executed. Absent the nth constraint, the system has a single-degree-of-freedom null space within whichmaneuvers can take place with minimum energy. Since far more degrees of freedom are necessary for usefultasks, the control-design problem is to find the path that minimizes energy subject to a weighted combinationof these constraints without necessarily satisfying them all exactly.

An alternative view of these constraints is that they provide a way to specify the central body angularvelocity ωB/N so that it minimizes the power for arbitrary joint velocities. This angular-velocity vector isregulated by the attitude-control system for the MaintenanceBot, and the operations concept may allow theuse of the central body for energy-minimization. We emphasize, however, that this approach is efficient only

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if the power required to steer the central body in this fashion is less than what such steering saves in thejoints.

To develop this law we coordinatize the vectors such that the projection of some arbitrary vector v ontoeach of a set of basis vectors C is written as the 3 × 1 matrixCv, i.e.,

Cv =[

c1 · v c2 · v c3 · v]

(8)

For example, these basis vectors may conveniently describe the orientation of the central body relative to anobject in the workspace to be manipulated. In any case, the central body angular velocity in C coordinatesCωB/N can be specified in terms of arbitrary joint angular rates in a way that minimizes power as follows:

CωB/N =

−C gTCn hrn

−C gTCn−1hrn−1

...

−C gTC1 hr1

‖hrn‖ sinφnC gn ·

(

Chrn ×C an−1

)

C gn ·(

Chrn ×C an−2

)

. . . C gn ·(

Chrn ×C a1

)

‖hrn−1‖ sinφn−1 0 C gn ·(

Chrn ×C an−2

)

. . . C gn−1 ·(

Chrn−1 ×C a1

)

......

. . .. . .

...

‖hr1‖ sinφ1 0 . . . 0 C g2 ·(

Chr2 ×C a1

)

θn

θn−1

...

θ1

(9)Where the superscript × indicates the skew-symmetric matrix equivalent of the cross product operation, andthe superscript + denotes the Moore-Penrose pseudoinverse. The angular-velocity vectors representation inC includes only three parameters; therefore, a robot with more than three links cannot experience minimum-energy dynamics. Instead, the pseudoinverse used in this expression provides a least-squares best CωB/N

given the possibly conflicting constraints. Again, the controls architecture may instead choose to weight thisconstraint for energy minimization relative to some other objective in defining the steering commands.

We emphasize that the singularity in this matrix is not directly related to the kinematic singularitiesassociated with robotic systems, whereby certain joints align in a way that would demand unrealizableactuator forces. Instead, when the power mapping discussed here becomes singular, certain joints simplycannot be used to minimize power. The question of kinematic singularities is an interesting and relevant one,but it is the same for the MaintenanceBot as for any other robotic system. The same design principles ofredundant joint degrees of freedom and singularity-avoidance apply here. The benefit of the MaintenanceBotsarchitecture is the use of CMGs as a ready means of limiting power but producing very high agility.

C. Power Comparison

Here we provide a simple example that compares the low-power, high-agility features of the MaintenanceBotto other approaches. In this example, the MaintenanceBot consists of a central body and a three-link arm.For simplicity, each links mass center is on its joint axis. Furthermore, the mass center of any system ofoutboard joints lies on the axis of the inboard joint, with the result that forces and mass-center motions areirrelevant. This principle is a worthy design goal, albeit difficult to achieve in practice (particularly for timeswhen the manipulator is carrying a payload). Nevertheless, we argue that this simplification helps make thepower comparison clearer for the sake of illustrating the concepts capabilities and is therefore justified in thecontext of this paper. For the same reason, products of inertia are taken to be zero. Other parameters in theexample are listed in Table 2. The design on which these parameters are based is somewhat arbitrary; butit corresponds roughly to a system of about 1m in characteristic link length (about a 2 m radius workspace)and no more than 100 kg overall mass.

In this example we consider three architectures. To allow a fair comparison, we require that the busremain inertially fixed in each case: i.e., any reaction torques from the robotic arms must be taken out bythe base-body attitude control system (ACS).

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Table 2. Parameters for the Two-Link Example

Parameter Value

Central Body Inertia 50 kg/m2 in all axes

Link 1 inertia 10 kg/m2 in all axes



CMG rotor momentum 50 Nms (Honeywell M50 CMG)

Link 1 CMG gimbal axes [0 1 0] in Link 1 coordinates



RWA rotor inertia 0.08 kg/m2 in all axes

CMG gimbal inertia 0.02 kg/m2 in all axes

CMG rotor rate (constant) 6000 RPM

Reference Configuration

Link 1 joint axis [1 0 0] in central-body coordinates



End-effector location [0 1 0] in central-body coordinates relative to Link 2 location

Initial CMG pair 1 gimbal angle 0 (momentum aligned with Link 1 joint axis)



1. The first case is what we have described as the baseline MaintenanceBot architecture: reactionless,CMG-driven joints. Each joint includes a scissored pair of 25 Nms CMGs (for a total capacity of 50Nms). It turns out that the base body does not move (regardless of the motion of the arm), so thebase-body ACS design is irrelevant here.

2. The second case is identical except that each scissored pair of CMGs is replaced by a 50 Nms reactionwheel (RWA). Once again, the base-body ACS is irrelevant.

3. The third case includes traditional direct- or geared-drive motors that actuate the joints. Reactiontorques applied the base body are significant, and they are compensated by a high-bandwidth reaction-wheel based ACS. The ACS uses four RWAs that are identical to those on the joints and whose spin

axes are[

± 12± 1

2

√2

2

]

The joints kinematics are varied numerically across a wide range in an effort to capture the worst-casepower and nominal statistics. This variation forms a Monte Carlo analysis, where the joint angles, angularrate, angular acceleration, and angular jerk are given values within the limits shown in Table 3. We takethese agility requirements to specify the joint kinematics, not the inertial kinematics. For example, theangular-rate limits apply to the joint, not the angular velocity magnitude of the link in an inertial frame.Thus, the end-effector agility is greater than that of a single link, up to twice the level shown in the table(e.g. 229 deg/sec), making the MaintenanceBot an extremely capable system. Another important pointis that the CMG gimbal-motor control-loop bandwidth is taken to be much higher than the characteristicfrequencies in the kinematics (e.g. the 2 rad/sec3 jerk), which is a reasonable assumption. This point allowsus to make the approximation that the CMGs achieve the prescribed kinematics instantaneously.

In this analysis the joint angles are always given uniform distributions. However, for rate, acceleration,and jerk two types of distributions are considered: a uniform distribution, meant to represent something likea day in the life of the MaintenanceBot, a statistically representative selection of maneuvers; and a simplemaximum or minimum, used to identify the worst-case power across all joint configurations. In all cases,we report only the power required if the electromechanical systems involved were lossless. In fact, someadditional multiplier (say 50%) should be added to account for various IR2 losses in harness and frictionlosses in bearings. This scale may not be quite the same for all systems but, it turns out, the savings are sogreat for the baseline architecture that the difference could not change the outcome of a trade study.

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Table 3. Agility Requirements for the MaintenanceBot Monte Carlo Example

Parameter Minimum Value Maximum Value

Joint angle (3 independent values) -

Joint rate (3 independent values) -2 rad/sec 2 rad/sec

Joint acceleration (3 independent values) -2 rad/sec2 2 rad/sec2

Joint jerk (3 independent values) -2 rad/sec3 rad/sec3

One rarely encounters a specification of jerk in this context. For the MaintenanceBot, there are severalimportant reasons for such a requirement. Among them, jerk is a direct measure of the changing acceleration,and thus the frequency content of the loading on both the MaintenanceBot and its payload. Minimizingjerk reduces the systems engineering effort in managing structural stiffnesses in the design and consideringfrequency-dependent coupled loads. However, requiring low jerk limits the bandwidth of the joint control.Jerk also limits the range of simultaneous rate and acceleration that can be achieved. For example, themaximum positive joint rate and the maximum positive joint acceleration cannot be applied simultaneouslyif there is a jerk limit because the acceleration must change over a finite time; and the rate limit would beexceeded during the period of negative jerk. Finally, the jerk is also directly related to the CMG gimbalacceleration. So, we have selected a jerk limit that seems to balance the demands of structural dynamicswith the desire for high-speed joint kinematics. We emphasize that it is the methodology that is of greatestinterest here; the specific values will be refined when MaintenanceBot parameters are allocated from higher-level performance requirements.

In all cases we assume that there is no regenerative power transfer; that is, the power bus must bedesigned to handle a current load equivalent to the absolute value of the mechanical kinetic-energy change(and losses). One might develop a cross-strapped set of actuators, in which power required by one is extractedfrom another. However, one of the goals of the MaintenanceBot is to use comparatively high-TRL solutions.And because CMGs and reaction wheels have flown very successfully for decades, we propose no significantmodifications to the Honeywell designs.

In case 1, the scissored-pair kinematics ( φ and its derivatives) are found from momentum conservationfor the kinematic chain, using the randomly selected kinematics from the Monte Carlo simulation:

I3·3ω

3/N

+ω3/N × I3 · ω3/N = τ3

I2·2ω

2/N

+ω2/N × I2 · ω2/N = τ2 − τ3

I1·1ω

1/N

+ω1/N × I1 · ω1/N = τ1 − τ2

IB ·Bω

B/N

+ωB/N × IB · ωB/N = −τ1

(10)

where B indicates parameters for the base body. However, with a non-moving base body (per our assump-tions), we have

I1·1ω

1/N

+ω1/N × I1 · ω1/N = −τ2

0 = −τ1

(11)

The jerk for a given body indicates how these torque changes in time (Nτ 3 ); e.g. for the third link,

I3·33ω

3/N

+2ω3/N × I3 · ω3/N + ω3/N ×

(

ω3/N × I3 · ω3/N

)

=Nτ 3 (12)

The momentum in the ith scissored pair is

hri = 2hr sin φ1ai (13)

Solving for momentum along the ai axis yields φi = sin−1 hri

2hr

. The torque it applies along the ai axis is

hri = 2hr cosφφi → φi =hri

2hr cosφi(14)

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and the time-varying torque (due to jerk) is

hri = 2hr

(

cosφφi − sinφiφi

)

→ φi =hri

2hr cosφi+ tanφiφi (15)

From these parameters, the power required by the ith CMG is

Pi =(

Ωrri + φigi + ωi/N)

· Ir ·

(

φigi+1ω

1/N

−Ωr ri × φigi −(

Ωrri + φigi

)

× ωi/N

)

+

(

φigi + ωi/N)

· Ig ·

(

φigi+1ω

1/N

−φigi × ωi/N

) (16)

where Ωr is the CMG rotor rate, ri is the rotor spin axis, φi is the gimbal rate, ωi/N is the ith links angular

velocity in an inertial frame N , Ir is the rotor inertia dyadic, φi is the gimbal acceleration,iω

i/N

is theangular acceleration of the ith link in an i-fixed frame, and Ig is the gimbal inertia dyadic.

The power is considerably simpler in the case of the reactionless RWA architecture. Beginning with theangular momentum for each link, we simply compute

hri = ai · Ii · ωi/N (17)

Similarly, the RWA torque is simply

τri = ai ·

(

Ii·iω

i/N

+ωi/N × Ii · ωi/N

)

(18)

The power for each RWA is then

Pi = τrihri

Ir(19)

where we have taken advantage of the fact that the rotor inertia is the same in all axes (by the assumptionsexplained above).

In the third case, we compute the time rate of change in the three-link systems kinetic energy Ejoints andtake that power to be required of the joint motors, whatever they may be. Then, the net momentum of theoutboard joints must be absorbed by the ACS RWAs (assuming no external torques on the MaintenanceBot),as must the torque of the three link system transmitted via the inboard joint:

P = Ejoints + EACS =

3∑

i=1

ωi/N · Ii·iω

i/N

+EACS (20)

The power due to ACS activity is then simply the shaft power for each of the four RWAs. Given the matrixA of spin axes (as defined in Table 1),

A =

0.5 −0.5 0.5 −0.5

0.5 0.5 −0.5 −0.5√2

2

√2

2

√2

2

√2

2

(21)

the four RWAs angular momenta hacs,i may be found from the base-body angular momentum hbase via

hacs,1

hacs,2

hacs,3

hacs,4

= AT(

AAT)−1

hbase (22)

The same result applies to reaction-wheel torques τacs,i Some straightforward algebra leads to

EACS =1

Ir|τ2

base|(

AAT)

|hbase| (23)

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where the absolute values (not the vector norms) are taken for each element because of our requirement thatthere is no flywheel-style energy storage, and that all power must come from the base body power bus or beshunted into a resistor.

The results of the Monte Carlo analysis for day-in-the-life statistics are shown in Figure 8. The firstcase, the baseline MaintenanceBot, requires about 151 W for typical operations, despite its extraordinarilyhigh agility. In contrast, the reactionless RWA case requires about 3300 W, and the traditional joint-drivearchitecture requires about 7200W.

Figure 8. Day in the Life Probability Density for Electromechanical Power: Three Cases, Both Linear andLogarithmic Scales Shown

The worst-case analysis (maximum rate, acceleration, and jerk, for all values of joint angle) reveals asimilar trend. The bar graph in Figure 9 shows that the CMG-based reactionless architecture requiresbetween 200 W and 1500 W for worst-case kinematics over the range of joint angles, while the RWA-basedarchitecture requires up to 70kW. The traditional joint-drive architecture demands an astounding 118 kW.

The obvious conclusion is that the CMG-based MaintenanceBot architecture can radically outperformother systems in power for high-agility maneuvers. Furthermore, this architecture opens up a large tradespace for power vs. agility, allowing highly effective MaintenanceBots to be incorporated for relatively lowpower-specific mass. Or, for a given mass, the MaintenanceBot can withstand more demanding operationsfor longer than competing architectures.

V. CMG Implementation Issues

A CMGs output torque is proportional to its rotor angular momentum and its gimbal rate. In thisapplication, the joint rate capability depends on this stored momentum, and the joint acceleration dependson both the CMG’s momentum and its gimbal-rate capability. High-speed rotors would therefore seem tosolve many problems: for a given momentum, the rotor would weigh less; or for a given rotor mass, themomentum would be greater. Figure 10 shows the influence of rotor size on momentum-storage capabilityand mass efficiency (momentum per kg). The strawman rotor design considered here consists of a 1 cmdeep x 3 cm high rim and a 0.5 cm thick web. This calculation does not account for the additional mass ofstructure, bearings, motors, and electronics associated with larger rotors.

Energy storage5 applications have investigated high-speed rotors for some time, although not for gimbaledapplications, where bearing loads are an issue. In the case of energy storage, material stress limits typicallyconstrain the rotor size. In space applications, however, design robustness demands careful attention tobearing life. In fact, it is bearing issues that have been partly responsible for the International Space Station

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Figure 9. Worst-Case Power Required (Kinematic Limits, All Joint Angles)

CMG failures.6 Honeywell’s history of successful design and operation of momentum systems for space isbased on robust bearing designs for rotor rates from 4500 to 6500 RPM.7 Although high-speed bearingtechnology will doubtless continue to advance for large rotors, the MaintenanceBot is designed for high TRLand therefore incorporates mature designs based on this approach.

An alternative approach is to implement extremely small rotors. Similar to mechanical directional gyros,which often use 400 Hz aircraft bus power to spin AC motors at a little less than 24,000 RPM, such devicesstand some chance of providing adequate momentum for compact robotic systems. However, bearing issueswould have to be addressed here, too, because directional gyros typically are not meant to transfer load acrossthe spin bearings.8 This approach was taken in the Manned Maneuvering Unit (MMU), which incorporatedthree scissored pairs of CMGs developed by David Osterberg of Honeywell Defense and Space ElectronicsSystems (Sperry at that time). Figure 11 shows a layout in the MMU. The three oblong boxes contain themodified directional gyroscopes.

Figure 10. Effect of Rotor Radius on Actuator Momentum Capacity and Mass

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Figure 11. Astronaut backpack with Three Scissored Pairs of CMGs

VI. Attitude and Arm Dynamics

The MaintenanceBot has several modes of operation:

1. Free flying with the arms fixed

2. Free flying with the arms moving

3. Free flying with one arm grasping the manned spacecraft

4. Fixed to the manned spacecraft via the grappling arm with the other arm free

5. Fixed to the manned spacecraft via the grappling arm with the other arm attached to the mannedspacecraft

The model in case 1 is for a rigid body with the CMGs. Considering a single CMG, the equation for themomentum is

H = A(Iω + Bhcmg) (24)

where hcmg is the vector momentum of the CMG in the CMG frame. B is a transformation matrix thattransforms from the CMG frame to the core body frame. A transforms from the body frame to the inertialframe. I is the total vehicle inertia and is assumed constant, even with the CMG rotation. The dynamicalequations are then

T = Iω + Bucmghcmg + ω×(Iω + Bucmghcmg) (25)

where hcmg is assumed constant ucmg is the unit CMG vector and T is the external torque due to thrusterfirings. Interestingly, if all of the CMGs have the same momentum vector we can generalize this equation toany number of CMGs by redefining B as the matrix sum of all of the transformation matrices for the CMGs.The derivative of the transformation matrix is

B = BΩ× (26)

where

Ω× =

0 −Ωx Ωy

Ωx 0 −Ωz

Ωy Ωz 0

(27)

which is known as the skew symmetric matrix for Ω. It is convenient to define the transformation matrix asthe product of a fixed transformation matrix and another matrix in which the rotation is only about x and

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the CMG momentum is around z. Therefore the derivative term becomes

Bucmghcmg = BF

0

− cos θ

− sin θ

θhcmg (28)

and the momentum term becomes

B = BF

0

− sin θ

cos θ

hcmg (29)

VII. Relative Sensing

A. Introduction

The MaintenanceBot uses two systems for relative attitude and position sensing. The primary system isan RF system that measures range and range rate from the MaintenanceBot to the manned spacecraft andcomputes relative attitude and position from those measurements. The manned spacecraft has 8 transmittersarranged on booms for this purpose. The RF system uses the same antennas and receivers that are used forRF communications. The RF system employs the ITT Low Power Transceiver.9

The second system is a single camera vision system which captures an image of the manned spacecraftand compares it against an image generated onboard the MaintenanceBot from a computer model of themanned spacecraft. This allows fine pointing and position control near the manned spacecraft. Since theMaintenanceBot always knows its position and attitude from the RF system the vision system need onlymake small corrections to the known state. The baseline camera is the Malin MSSS.10

The vision relative attitude and position system is illustrated in figure 12.

K a l m a n F i l t e rR a n g e , R a n g eR a t eV i s i o n S y s t e m M a n e u v e rP l a n n e r O p t i m a l L a m b e r tF o r m a t i o n F l y i n gA * C l o s eM a n e u v e rR FA t t i t u d e S e n s o r sFigure 12. Block diagram of the relative navigation and maneuver planning system

B. RF

The RF system provide the primary navigation sensing of the MaintenanceBots relative to the mannedspacecraft. To achieve sufficient position dilution of precision (PDOP), generally taken to be less than 6, theRF system must have a sufficient antenna baseline to cover desired orbital distances and enough antennasto assure a solution at any location. The geometric dilution of precision for pseudorange measurements isdefined from the linearized navigation solution from at least 4 pseudorange residuals.

GDOP =√

trace[HT H ]−1 (30)

where the rows of H are defined using the directional cosines from the MaintenanceBot to each antenna,

hi = [ ui −1 ] (31)

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The position dilution is computed by summing only the position elements in the trace operation, excludingthe time dilution.

Consider a system of six antennas, arranged in two planes, each plane an equilateral triangle, as shownin Figure 13.

−100

−50

0

50

100

−40−20020406080

−50

0

50

x

y

z

Figure 13. Example RF antenna arrangement and relative orbit for PDOP calculations

The plot on the left side of Figure 14 shows the PDOP for half an orbit around the center of the system.Taking a single point on this orbit, the plot on the right shows the decrease in PDOP with radial distance.In general, the MaintenanceBots can get good solutions at radial distances equal to and somewhat largerthan the antenna baseline.

PDOP with Orbital Radius for Antenna Baseline of 100 m

0 0.5 1 1.5 2 2.5 3 3.50

5

10

15

20

25

PD

OP

Angle (rad)

50

100

200

PDOP Variation with Baseline and Distance

0 50 100 150 200 250 3000

5

10

15

20

25

30

PD

OP

Radial distance (m)

50100150

Figure 14. PDOP results for example configuration

C. Vision

At long distance, when features of the target are not available, the sensor outputs range rate, range, azimuthand elevation from a ladar. These are input to an iterated extended continuous discrete Kalman Filter whichestimates both the relative state and the absolute state of the manned vehicle. The manned vehicle orbitis determined from range and range rate measurements from ground stations (using standard multiple toneranging) or from navigation satellites. Currently navigation satellites are only available around the earth butit is expected that navigation satellites will be placed in orbit around the Moon and Mars to support future

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missions. When the two vehicles are closer the CAD based algorithm returns relative position supplementedby range measurements until the minimum range is reached.

The relative attitude determination system incorporates manned vehicle gyro measurements and anyexternal measurements (e.g. star tracker, RF, etc.) to produce a best estimate of the relative attitude.

The vision system is illustrated in figure 15. It has been partially prototyped in MATLAB, and thegeneral concept of optimal image matching demonstrated to within the capability of MATLAB’s imagesoftware and the optimization method used.

C A D M o d e l I l l u m i n a t i o n M o d e lR e n d e r i n gT r a n s f o r m a t i o n F i l t e r i n gC a m e r a C a m e r aP r o c e s s i n g F i l t e r i n g

C e n t r o i d i n gC e n t r o i d i n g

F i n eM e a s u r e m e n tR o u g hM e a s u r e m e n t AK a l m a n F i l t e r

K a l m a n F i l t e rFigure 15. Vision System

An onboard CAD model generates an internal representation of the manned vehicle. Centroiding isapplied to this model and the image in the camera to get a rough position estimate used to start the iterativeprocess. A range measurement is used if available. The CAD and camera image are filtered. Filtering isdesigned to enhance the image to speed convergence of the optimization algorithm. Filtering may includea filtering (e.g. Gaussian), edge detection and point detection algorithms. The filtered images (which mayinclude color images, edges or points) are processed to get the attitude and position. If sufficient points havebeen identified a point algorithm can be used to determine the position and orientation directly. The resultingposition and orientation is fed to a Kalman filter which maintains the state of both the MaintenanceBot andmanned vehicle.

VIII. Close Maneuvering

Close maneuvering is defined as orbiting the target vehicle, thrusting continually in the radial direction.This is more fuel efficient than forming an equilateral triangle and performing delta-Vs at the apexes, andit provides a constant circumnavigation radius.

The dynamical equations for planar motion, written in cylindrical coordinates are

r − rθ2 = ar (32)

rθ + 2rθ = aθ (33)

This plane has the target at the center. Perturbing forces are ignored. The control equations are

ar = a sin γ (34)

aθ = a cos γ (35)

where γ is the angle between the thrust vector and the orbit tangent and a is the magnitude of the acceler-ation. The desired accelerations are

ar = −rθ2 − kr (36)

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andaθ = rθc (37)

The constant gain k provides damping of radial velocity, which should be zero.The control law assumes that the vehicle starts at the desired distance from the target. The actual

control parameters are the magnitude of the control acceleration

a = r

√

a2r + a2

θ (38)

and the steering angle

γ = atan(ar

aθ) (39)

A simulation is shown in figure 16. The rightmost plot shows the state variables. The radius settles toa slight offset since the controller does not feedback the radius. Small errors in radius are acceptable. Theradial rate is damped to zero very quickly. The circumferential angular velocity reaches a constant and theangle increases linearly. No attempt is made to correct the circumferential rate for perturbative accelerations.Initially the acceleration magnitude is large to accelerate the vehicle to the desired circumferential rate. Itthen drops to the magnitude required to maintain the radial position. The pointing angle decreases slowlythen jumps to -90 degrees when the desired circumferential angular velocity is achieved.

State

20

20.01

r

0

0.5

1

θ

0

0.5

1x 10

-3

r do

t

0 10 20 30 40 50 60 70 80 90 1000

0.005

0.01

0.015

θ do

t

Time (sec)

Control

0

0.005

0.01

0.015

0.02

0.025

a

0 10 20 30 40 50 60 70 80 90 100-2

-1.5

-1

-0.5

0

γ

Time (sec)

Figure 16. Close maneuver acquisition simulation

An analysis script generates the radial force required for circumnavigations of varying radii and periods.

Radial Force

0 10 20 30 40 50 60 70 80 90 10010

-3

10-2

10-1

100

101

102

Rad

ial F

orce

(N

)

Orbit Radius (m)

P = 1 minP = 5 minP = 10 minP = 15 minP = 20 min

Figure 17. Radial Force

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The A* search algorithm can be used to deviate from a constant radius to avoid stayout zones. Thiswould be necessary during an inspection of the manned spacecraft since the MaintenanceBot would want tostay out of the sensor fields-of-view, RF transmitter booms, etc. A* searches in a state space, in this case agrid around the target at the desired radius, for the least costly path from a start state to a goal state. Thecost during a maneuver is the change in angle needed to go from one state to another. A* is guaranteedto find the shortest path as long as the heuristic estimate of the path cost for each node is admissible -that is, never greater than the true remaining distance to the goal. It makes the most efficient use of theheuristic function so that no search that uses the same function and finds optimal paths will expand fewernodes than A*.11 The algorithm requires two lists of states called open and closed for unexamined andexamined states. At the start closed is empty and open has only the starting state. In each iteration thealgorithm removes the most promising state from open for examination. If the state is not a goal then theneighboring locations are sorted. If they are new they are placed in open. If they are already in open theinformation about the states is updated if this is a cheaper path to those states. States that are already inclosed are ignored. If open becomes empty before the goal state is reached then there is no solution. Themost promising state in open is the location with the lowest cost path through that location. This heuristicsearch ranks each node by an estimate of the best route that goes through that node. The typical formulais expressed as:

f(n) = g(n) + h(n) (40)

where: f(n) is the score assigned to node n, g(n) is the actual cheapest cost of arriving at n from the startand h(n) is the heuristic estimate of the cost to the goal from n In this problem the state space is a set ofpoints on a sphere and the cost is the change in angle needed to go from one state to another.

To circle the target it is necessary to run A* three times with points chosen at two intermediate locationson the sphere. Figure 18 shows the path. Each color is a segment of the path. As can be seen the selectionof grid points is critical as it determines the actual path. Denser grids require more computation but wouldresult in smoother paths. The ideal grid would have more points around obstacles and fewer points in openareas. There is also nothing the prevent the use of 3-dimensional grids if it is desired to change the radiusof the orbit around the target vehicle.

-0.50

0.5

-0.5

0

0.5

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

x

A* Path

y

z

Figure 18. A* path

The A* planner generates a trajectory rc. This is passed through a first order digital filter

rci = mrci + (1 − m)rc (41)

where m is between 0 and 1. 0 is all-pass and 1 is no-pass. The error passed to the controller is the differencebetween the measured relative state and the output of the filter. This serves to smooth the inputs to thecontroller thus smoothing out the abrupt direction changes from A*.

u = xm − rci (42)

The controller is a proportional derivative controller with rate filtering. The commanded acceleration is

y = cxk + du (43)

20 of 31


and the propagation of the controller state is

xk+1 = axk + bu (44)

The commanded force isf = −my (45)

More precise tracking is possible using a PID (proportional integral differential) controller but would onlybe necessary when maneuvering around targets with very tight maneuver tolerances.

The left-hand side of figure 19 shows the accumulated delta-v during the maneuver. It is 23% higherthan a maneuver without a stayout zone. If there are no stayout zones the A* maneuver requires the samedelta-v as an ideal circular maneuver with constant radial thrusting. This is despite the fact that the costdoes not explicitly involve velocity change.

Tracking performance is shown on the right-hand side of figure 19. The controller smoothes out theabrupt changes in the trajectory requested by A*. The abrupt changes are due largely to the grid. A PIDor higher gain controller would track the trajectory more precisely at the cost of more fuel consumption.

Delta−V

0 1 2 3 4 5 60

2

4

6

8

10

12

14

16

18

20

DV

, km

/s

Time (min)

Maneuver Tracking

-100

0

100

200

x

20

30

40

50

60

y

0 1 2 3 4 5 6-200

-100

0

100

200

z

Time (min)

ActualCommanded

Figure 19. Tracking Delta-V and Performance

IX. Formation Flying

Formation flying of spacecraft is an enabling technology for a wide range of missions, including the mul-tiple MaintenanceBotconcept. In general, formation flying involves actively controlling the relative positionand velocity of two or more spacecraft to establish a desired geometry. Examples include space-based inter-ferometery and synthetic aperture radar, each of which utilize large separation distances between spacecraftto conduct high-resolution sensing.

For the majority of the time on-orbit, each MaintenanceBotwill fly in formation with the manned space-craft. This mode of flight will be interrupted only in two cases: 1) when it docks, to refuel and and recharge,and 2) when it performs a servicing task. The primary reason for flying the MaintenanceBotsin formation isso that they may provide continuous visual inspection. An additional motivation is to enable a more rapidresponse to emerging events that require immediate inspection and/or service.

In order to minimize fuel-usage, the MaintenanceBotswill follow natural repeating trajectories. Theseare closed trajectories relative to the prime spacecraft that repeat each orbit period and, in the absence ofdisturbances, require no applied force to maintain. Relative motion is defined within the Hill’s coordinateframe – a non-inertial, rotating frame where the origin is fixed to the manned vehicle. Figure 20 shows therelative frame, hereafter referred to as Hill’s frame. The x-axis points in the zenith direction, the z-axisis aligned with the angular momentum vector, normal to the orbital plane, and the y-axis completes theright-hand system. For circular orbits, y is always aligned with the velocity vector.

Whether the manned spacecraft is in a circular or eccentric orbit, families of repeating, periodic relativetrajectories can be readily found. The constraint for achieving periodic relative motion with two satellites is

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z xy ν p e r i g e e

Figure 20. Relative Hill’s Frame. The Origin is Centered at the Manned Spacecraft

that their semi-major axes be equal. A small difference in mean anomaly will produce an along-track offset,giving a leader-follower type of formation. By introducing differences in eccentricity and/or the argumentof perigee, relative motion in the orbit plane can be achieved. In circular orbits, this in-plane motion issometimes referred to as a football orbit, because the trajectory takes the shape of a 2 × 1 ellipse. Finally,cross-track motion, which is decoupled from in-plane motion, can be created through differences in the rightascension and/or inclination. The geometry of the relative motion in a circular is easily expressed as asuperposition of along-track offset, in-plane elliptical motion, and cross-track oscillation. The following fiveparameters are used to fully define the geometry of any type of relative trajectory with a circular referenceorbit:

Table 4. Geometric Parameters for Relative Motion in Circular Orbits

Parameter Description

y0 Along-track offset. Defines the center of the in-plane relative ellipse.

aE

Semi-major axis of relative ellipse.

β0 Phase angle on relative ellipse at ascending equator crossing.

Measured positively from −x axis to +y axis of Hill’s frame.

zi Cross-track amplitude due to inclination difference.

zΩ Cross-track amplitude due to right ascension difference.

An example trajectory is shown in Figure 21, illustrating the separate in-plane and out-of-plane motions.

Figure 21. Example of Relative Motion in Circular Orbits

The relative motion occurring in eccentric orbits is fundamentally different from that found in circular

22 of 31


orbits. The in-plane motion no longer follows a 2×1 ellipse, and the cross-track oscillation is not necessarilycentered about the origin. The shape of the trajectory depends upon the eccentricity, as well as the pointsin the orbit where the maximum radial and cross-track amplitudes occur. The geometry of the relativetrajectory can be defined using the parameters in Table 5.

Table 5. Geometric Parameters for Relative Motion in Eccentric Orbits

Parameter Description

y0 Center of along-track motion

x Maximum radial amplitude

νx True anomaly where x occurs

z Maximum cross-track amplitude

νz True anomaly where z occurs

For example, consider a reference orbit with an eccentricity of 0.7. The along-track motion is centeredat 1.0 km, and a maximum radial amplitude of 1.0 km occurs at ν = 90 deg. In addition, a maximumcross-track amplitude of 1.0 km occurs at ν = 180 deg. The trajectory is shown in Figure 22. The greenshaded regions illustrate the projection of the motion onto the x-z and x-y planes.

−2−1

01

23

−2−1

01

23

−2

−1

0

1

2

x [km]Radial

z_

y0

x_

y [km]Along−Track

z [km]Cross−Track

Figure 22. Example Relative Trajectory in an Eccentric Orbit

Now consider a different example, where the reference orbit has an eccentricity of 0.6. The along-trackmotion is still centered at 1.0 km, and the maximum radial amplitude of 1.0 km now occurs at ν = −90 deg.Also, the maximum cross-track amplitude of 1.0 km now occurs at ν = 127 deg (which corresponds to aneccentric anomaly of E = 90 deg). This trajectory is shown in Figure 23 on the following page. Note theconsiderable difference in the motion when projected onto x-z and x-y planes. The symmetric figure-eightshape in the x-z plane is achieved by forcing z to occur at E = 90 deg.

For a given number of MaintenanceBotsflying in formation with the manned vehicle, we wish to designa set of relative trajectories that meet the following objectives:

1. Maintain a minimum “safe” separation distance between the manned spacecraft and all other Mainte-nanceBots at all times.

2. Avoid interference with the manned spacecraft’s ground communication and other sensing payloads.

3. Provide maximum visual coverage over the exterior surface of the manned spacecraft.

4. Minimize differential disturbances that lead to increased delta-v requirements to maintain formation.

These objectives provide a series of specific geometric constraints that must be observed when choosing theparameter set for each trajectory.

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R[htb]

−2

0

2

−4

−2

0

2

4

−3

−2

−1

0

1

2

x [km]Radial

y0

z_

x_

y [km]Along−Track

z [km]Cross−Track

Figure 23. Example Relative Trajectory in an Eccentric Orbit

Given enough MaintenanceBots, the relative trajectories can be designed so that the entire surface of themanned vehicle is imaged once per orbit. This requires a combination of in-plane and cross-track motion thatputs the MaintenanceBots above, below, and to the side of the manned vehicle at different points throughoutthe orbit. An example formation of four MaintenanceBots is shown in Figure 24. An eccentricity of 0.1 isused. The minimum separation distance between any two MaintenanceBotsis 120 meters, and the closestthat any MaintenanceBotcomes to the manned vehicle is 50 meters.

−0.1

0

0.1

−0.2

−0.1

0

0.1

0.2

−0.1

−0.05

0

0.05

0.1

x [km]

y [km]

z [km]

Figure 24. Example Formation to Provide Complete Visual Coverage

The first three items listed above are specific to the manned spacecraft design, its size and features, aswell as the number of MaintenanceBots. The fourth item, however, concerns the nature of relative orbitdynamics, and depends upon the orbit. The most significant relative disturbances that will impact formationflying performance are: differential drag, differential solar force, and the effects of gravitational perturbations.The J2 perturbation tends to cause significant secular drift when an inclination difference is present. Theinduced drift-rate is largest in LEO, and drops off quickly as the semi-major axis is increased. In LEO,the largest relative disturbance is likely to be differential drag, since the size of the manned vehicle will bemuch greater than that of the MaintenanceBots. It is possible, however, to intentionally design a relativetrajectory with inclination difference so that the drift from J2 serves to partially counteract the differentialdrag.

The MaintenanceBots must maintain their desired relative trajectories in the presence of these differentialdisturbances. Because formation flying will represent a significant portion of the time on-orbit, it is important

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to devise a formation maintenance strategy that requires as little delta-v as possible. The general approachis to define an allowable position deadband, or error-box, for each MaintenanceBot, so that as long as itstays within the box it does not maneuver. Once the position error exceeds this deadband, a correctivemaneuver is planned and implemented. This on-off control approach leads to a limit-cycle behavior, wherethe frequency depends upon the accuracy of the relative state estimate and the size and direction of therelative disturbances.

Linear programming (LP) is a particularly interesting technique for planning formation flying maneuvers.The application of LP methods to the problem of relative orbit control has been presented in various formsin recent years.12, 13, 14, 15 In general, the LP approach is used to compute an impulsive delta-v sequenceover a fixed time window, so that the desired state is reached at the final time and a given cost function isminimized. The cost function may be defined to simply represent the total delta-v for the maneuver, or itmay be made more complex, incorporating specific hardware-related or mission-related constraints.

The application of LP control requires that the relative dynamics be expressed as a linear system instate-space. The relative dynamics for a circular reference orbit are linear time-invariant (LTI), and may beexpressed as:

x(t) = Ax(t) + Bu(t) (46)

y(t) = Cx(t)

where x is the state vector, u is the control input, and y is the output. The state vector consists of the

relative position and velocity in Hill’s frame, x =[

x y z x y z

]⊤, the control input is the applied

acceleration in Hill’s frame, and the output is equal to the state vector (C is identity).The A and B matrices are taken directly from the linearized equations of motion. For a circular reference

orbit, the matrices are independent of time:

A =

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

3n2 0 0 0 2n 0

0 0 0 −2n 0 0

0 0 −n2 0 0 0

B =

0 0 0

0 0 0

0 0 0

1 0 0

0 1 0

0 0 1

(47)

where n is the mean orbit rate. The continuous-time system is discretized using a zero-order hold over atime-step ∆t to obtain:

xk+1 = Axk + Buk (48)

yk = xk

where A and B now denote the discrete-time state-space matrices. The expression for the second state canbe found as follows:

x2 = Ax1 + Bu1

= A (Ax0 + Bu0) + Bu1 (49)

= A2x0 + ABu0 + Bu1

Extending to the N th state, we have:

xN = ANx0 +[

AN−1B AN−2B · · · AB

]

×

u0

u1

...

uN−1

(50)

For future convenience, let us introduce the following definitions:

B =[

AN−1B AN−2B · · · AB

]

(51)

u =[

u0 u1 · · · uN−1

]⊤

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so thatxN = ANx0 + Bu (52)

This gives us an expression for the N th state in terms of the initial state, x0, and the control history, uk,for k = 0 → N − 1. The objective is to find a control history that requires the minimum cumulative delta-v,subject to the constraint that the desired state is achieved at k = N .

The terminal constraint may be written as:

|xN − x∗| ≤ ǫ (53)

for a sufficiently small ǫ vector. Noting that the right-hand side is an absolute value, we may rewrite theabove expression as two inequalities.

xN − x∗ ≥ −ǫ (54)

xN − x∗ ≤ +ǫ

Substituting Eq. (52) into Eq. (54), we obtain:

ANx0 + Bu − x∗ ≥ −ǫ (55)

ANx0 + Bu − x∗ ≤ +ǫ

Further algebraic manipulation yields:

− Bu ≤ ǫ + ANx0 − x∗ (56)

Bu ≤ ǫ − ANx0 − x∗

Now let A and b be defined as:

A =

[

−B

B

]

(57)

b =

[

ǫ + ANx0 − x∗

ǫ − ANx0 + x∗

]

so that the inequality may be written as follows:

Au ≤ b (58)

The problem is now posed in a form suitable for the Simplex algorithm, which is a well-known techniquefor solving LP problems. The objective is to minimize the cost cu subject to the constraint defined inEq. (58). Here, the cost coefficient vector, c, has 3N columns, and is nominally composed of all 1’s. Theweights may be adjusted to provide a greater or lesser penalty at different times, or for different axes inthe relative frame. The approach for setting up the LP problem for eccentric orbits is described in anotherpaper.16

X. Example Manned Spacecraft

For the purposes of this paper a conceptual design for a spacecraft for a Mars mission was designed. It is anuclear thermal vehicle with hydrogen fuel using the Small Engine Reactor developed by Los Alamos.17 Thevehicle is designed to put a 125,000 kg payload into low Mars orbit. It is assumed that the vehicle is refueledat Mars. Unlike nuclear electric vehicles, nearly all of the reactor power goes into the fuel so relatively littleradiator area is needed. The vehicle carries 6 crew members to Mars orbit where they descend to the surfacein the lander which uses direct thrust for landing. The lander would shuttle fuel back to the transfer vehicleduring its 500 day stay on Mars.

Figure 28 shows a computer generated image of the manned spacecraft.The dimensions in the figure are in meters. The large trusses support the RF transmitters for the local

navigation system. They also support the high gain antennas and other instruments. The crew quartersare a ring which spins to provide some artificial gravity. The Small Engine Reactor is seen on the left handside. The engine has a 2π steradian radiation shield so it must be approached from the nozzle end by theMaintenanceBot. The hydrogen fuel provides shielding, of varying thickness, during most of the mission. Thedelta-V budget is based on impulsive burns. This approximation is justified since the engine can consumeall of the fuel in less than a day.

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Table 6. Hypothetical Mars Transfer Vehicle (MTV)

Property Value

Transfer Vehicle

DV Earth Escape 3.15 km/s

DV Earth to Mars 5.59 km/s

DV Mars Capture 1.45 km/s

DV Total 10.20 km/s

Lander

DV to 15 km 3.76 km/s

DV 15 km to 100 km 0.04 km/s

DV Total 3.81 km/s

Lander Mass

Mass Dry 51,711 kg

Mass Payload 4,000 kg

Mass Fuel 73,024 kg

Mass Total 129,395 kg

Transfer Vehicle Mass

Mass Dry 124,066 kg

Mass Fuel 579,839 kg

Mass Payload 129,395 kg

Mass Total 833,960 kg

Thrust 72 kN

Exhaust Power 308 MW

XI. MaintenanceBot System Design

The MaintenanceBot has 24 cold gas thrusters, 4 SCMGs (3 orthogonal and 1 skew), a lithium ionbattery, dual triple junction solar panels and an ITT Low Power Transceiver (LPT) for communications andGPS processing. The LPT can handle up to 12 antennas. There are 8 GPS antennas and 4 omni antennasfor intervehicle links and links to the ground.

The docking apparatus is conceptually similar to the system developed by Prof. Michael Swartout18 andinvolves a cone which connects to a conical adaptor. The connectors are for high pressure N2 and power.All communications are via RF so no data links are required in the docking adaptor.

A preliminary model for the MaintenanceBot core is shown in the following figure.The scale is in meters. The MaintenanceBot has two arms. One has three links and is used for working

on the manned spacecraft. The other is a single link and is used to anchor the MaintenanceBot to themanned spacecraft while it is working. This allows the MaintenanceBot to unload momentum without firingthrusters.

Some of the major baseline components are given in Table 7.It is anticipated that there would be a bay on the manned spacecraft into which the MaintenanceBot

could fly. The bay would then be pressurized so the crew could repair MaintenanceBots. This would includereplacement of CMGs and other components. The ability to replace CMGs reduces the lifetime requirementson the bearings thus reducing the need for very sophisticated bearing assemblies. Since work would be doneon the MaintenanceBot inside the bay when it was pressurized, no special equipment would be needed forrepairs, other than what was available to technicians during Integration and Test.

XII. Simulation

A close orbit maneuver was simulated as an example. The control system commands the MaintenanceBotto maintain a steady attitude rate around the y-axis will the z-axis thruster fire to maintain a steady radial

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Figure 25. Mars Transfer Vehicle (MTV)

force. The CMG steering law uses simplex. Simplex solves the problem

U(θ)Ω = T (59)

where each column of U(θ) multiplied by the appropriate CMG rate is the torque produced by the CMGand T is the torque demand. Each column of U is a function of the current CMG angle. Each CMG isrepresented by two unit vectors in U with opposite signs so that both positive and negative rates can result.

Simplex weights the size of the CMG angle so that CMGs with large angles are used less than those withsmall angles. This is not the most sophisticated CMG steering law but can be implemented with only a fewlines of code.

The control algorithm is a rate controller with an additional term to cancel the Euler coupling of theCMGs.

Figure 28 shows a computer generated image of the MaintenanceBot in orbit around the manned vehicle.Figure 28 shows the body rates and CMG angles during the simulation. The rates keep the cameras on theMaintenanceBot pointing at the manned vehicles.

XIII. Conclusion

A preliminary design for a maintenance robot for manned vehicles is presented in this paper. It makes useof single gimbal control moment gyros for attitude control and robot arm movement and cold gas thrustersfor formation flying and close maneuvering. CMGs present a clear advantage over reaction wheels or thrustercontrol for maneuvering of the core body. Use of the CMGs for the robot arms is feasible with existing CMGsbut would potentially be enhanced by CMGs using higher speed rotors.

Future work will involve the incorporation of robot arm dynamics into the simulation and the developmentof algorithms (i.e. inverse kinematics) to position the arm. The various control systems described in thispaper will be integrated and integrated simulations of the system will be conducted. The crew interface willbe defined, developed and tested as part of the simulation. With regard to the CMGs studies will continueon developing more compact CMGs for arm actuation.

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Figure 26. MaintenanceBot

Table 7. MaintenanceBot components

Component Model Number Vendor Reference

SCMGS N/A 4 + 2 per manipulator dof Cornell

Integrated Avionics Unit RAD 750 1 Broad Reach/BAE Systems 19

Battery Lithium ion 1 Japan Storage Battery Ltd. 20

Gyro ADXRS150 3 Analog 21

Camera 1 TPS Malin 10

Thrusters 50-673 24 Moog 22

Comm 1 LPT ITT 9

Temperature Transducers 40 AD590 Analog Devices 23

Acknowledgments

The close maneuver algorithms discussed in this paper were developed under NASA Contract, H7.02-9628and Air Force Contract F29601-02-C-00. The formation flying algorithms are currently under developmentunder NASA contract NNG04CA08C.

References

1NASA/JSC, “Robonaut,” World Wide Web, http://vesuvius.jsc.nasa.gov/er er/html/robonaut/robonaut.html, 2004.2Robotics, M., “Dextre,” World Wide Web, http://www.space.gc.ca/asc/eng/media/press room/, 2004.3“Honeywell M50 Flysheet,” Tech. rep.4Quartararo, R., “Course Notes, Introduction to Spacecraft Attitude Control,” Tech. rep., 2003.5McLallin, K. L., Fausz, J., and et al., “Aerospace Flywheel Technology Development for IPACS Applications,” Jul 2001.6“NASA Press Release, STS-111,” Tech. Rep. 9, Jun 2002.7“Honeywell M50 Flysheet,” Tech. rep.8Peck, M., Miller, L., and et al., “An Airbearing-Based Testbed for Momentum Control Systems and Spacecraft Line of

Sight,” Feb 2003.9ITT, Low Power Transceiver (LPT), http://www.ittspace.com/products/lowpow.asp.

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Figure 27. MaintenanceBot in orbit around the manned vehicle

CMG Angles

-0.5

0

0.5

θ x

0

1

2

θ y

-1

-0.5

0

θ z

0 1 2 3 4 5 6 7 8 9-0.5

0

0.5

θ s

Time (min)

Body Rates

-2

-1

0

1x 10

-3

ωx

0

0.005

0.01

0.015

ωy

0 1 2 3 4 5 6 7 8 9-5

0

5

10

15x 10

-4

ωz

Time (min)

Figure 28. MaintenanceBot in orbit around the manned vehicle

10Systems, M. S. S., MSSS Offers Space-Qualified Camera Systems For All Mars Scout, Discovery, and New FrontiersMission Requirements, http://www.msss.com/camera info.

11Stout, W. B., The Basics of A* for Path Planning , Charles River Media., 1999.12Tillerson, M., Inalhan, G., and How, J., “Co-ordination and control of distributed spacecraft systems using convex

optimization techniques,” International Journal of Robust Nonlinear Control , Vol. 12, No. 1, 2002, pp. 207–242.13Breger, L., Ferguson, P., and How, J., “Distributed Control of Formation Flying Spacecraft Built on OA,” AIAA Guid-

ance, Navigation and Control Conference and Exhibit , Austin, TX, 2003.14Richards, A., Schouwenaars, T., How, J., and Feron, E., “Spacecraft Trajectory Planning with Avoidance Constraints

Using Mixed-Integer Linear Programming,” Journal of Guidance, Control, and Dynamics, Vol. 25, No. 4, 2002.15Tillerson, M. and How, J., “Formation Flying Control in Eccentric Orbits,” AIAA Guidance, Navigation and Control

Conference and Exhibit , Montreal, Canada, 2001.16Mueller, J. B., “A Multiple-Team Organization for Decentralized Guidance and Control of Formation Flying Spacecraft,”

No. AIAA-2004-6249, Sep.17Joseph A. Angelo, J. and Buden, D., Space Nuclear Power , Orbit Book Company, 1st ed., 1985.18Fitzpatrick, T., Satellite as small as a cantaloupe docks with mothership size of a medicine ball ,

http://mednews.wustl.edu/tips/page/normal/4147.html.19Broad-Reach, Integrated Avionics Unit , http://www.broad-reach.net/iau.html.20Ltd, J. S. B., Lithium ion battery , http://www.nippondenchi.co.jp/npd e/lithium/lithium.html.21Devices, A., iMEMS Gyroscopes, http://www.analog.com/Analog Root/sitePage/mainSectionHome.

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22Moog, Moog Cold Gas Thruster Systems, www.moog.com/Media/1/spdcat.pdf.23Devices, A., “Two Terminal IC Temperature Transducer AD590,” 1997.

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