Control-Moment Gyroscopes for Joint Actuation: A New ... - CiteSeerX

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 components of the spacecraft that are only accessible from the outside. This paper presents a design of a novel free-flying maintenance robot (known as a MaintenanceBot.) The MaintenanceBot uses Control Moment Gyros (CMGs) for manipulator arm and attitude control. This provides high authority control in a compact low power package. Relative position and attitude determination is accomplished with an RF system supplemented by a vision system at close range. When not docked to the manned vehicle (which must be done periodically to refuel and recharge batteries or when the manned vehicle performs orbit changes) the MaintenanceBots fly in formation using a cold gas thruster system and formation flying algorithms that permit dozens of MaintenanceBots to coordinate their positions. The use of CMGs is a prominent feature of this design. An array of CMGs can exchange angular momentum with the spacecraft body to effect attitude changes, as long as certain mathematical singularities in the actuator Jacobian are avoided. The proposed maintenance robot benefits dramatically from the dynamics and control of a multibody robotic arm whose joints are driven by CMGs. In addition to high power efficiency, another advantage of this concept is that spacecraft appendages actuated by CMGs can be considered reactionless, in the sense that careful manipulation of the CMG gimbal angles can virtually eliminate moments applied to the spacecraft body. This paper provides a preliminary design of the MaintenanceBot. Analysis of the formation flying and close maneuver control systems is included. Simulation results for a typical operation is provided.

I.

Introduction

ree flying maintenance robots have the potential to greatly increase the safety of future manned missions. F 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 systems developed at PSS and control-moment gyro (CMG) attitude and effector control in development at Cornell University. Because of their high agility and low power requirements, these robots will help improve system reliability by allowing in-situ repairs for which an astronaut would otherwise require extensive training and will be able to undertake tasks that would otherwise be too dangerous or require too much mechanical strength for an astronaut. Many robotic manipulators have been proposed for space applications. Robonaut1 developed at NASA’s Johnson Space Center in Houston, Texas, has five-fingered, multi-jointed hands. Ranger, developed at the University of Maryland in College Park has tools that plug into each wrist socket. Dextre,2 a product of MD Robotics in Brampton, Ontario, will be launched to the International Space Station in 2007. Dextre is a 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. 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.

† President,

1 of 31 American Institute of Aeronautics and Astronautics

high torque CMGs which reduce the need for anchoring allowing it to operate around any vehicle and at any location around the vehicle. Dextre is shown in Figure 1. Momentum unloading of the Single Gimbal Control Moment Gyros (SCMGs) is still most effectively accomplished by anchoring the MaintenanceBot to the spacecraft and unloading the SCMGs. The MaintenanceBots directly address several of the challenges that need to be met for sustained manned exploration: • They represent a radically low-power and high-redundancy means of in-space assembly and maintenance • Their sophisticated software and their low power ensures longduration autonomous operation • They are designed for reusability, to follow a larger spacecraft and perform servicing tasks on demand • They can be reconfigured for specific assembly tasks, merely by selecting different end effectors; furthermore, the scalability of this technology allows the MaintenanceBot software and missionoperations procedures to be reused in a modular fashion for multiple tasks

Figure 1. Dextre

• Multiple CMGs for attitude and manipulator actuation ensures system-wide mechanical and control redundancy; the networkcentric object-oriented flight-operation software ensures margin and redundancy 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 can offer up to 5 Nm per watt of electrical power. The novel, multiple-redundant use of CMGs for attitude and dexterous control distinguishes this concept from classical on-axis actuation of robotic arms, which leads to low torque per mass and low torque per Watt. High-agility servicing systems will be key to providing multipurpose, 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 system retains its promise of high torque for low power. Maturing this system will involve integrating these mid-TRL technologies (CMGs and agile formation flight software) into a flight-qualified system and demonstrating their operation 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 the flexibility needed for maneuvering around a large, irregularly shaped vehicle such as a lunar-transfer vehicle or 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 1 requirements were developed. These are presented in Table 1 on the following page. These provide a basis for the design of the MaintenanceBot in this paper. Requirement 1 is the minimum functionality required. This alone is a necessary function that would have proven invaluable on several manned space missions. Requirement 2 is the next step up in functionality. At this stage we assume that the components are designed for robotic or astronaut replacement. Requirements 3-7 define the operational capabilities of the MaintenanceBot. Requirement 7 permits the simultaneous operation of many MaintenanceBots. This requirement implies that the MaintenanceBots, when not under crew 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 requirement for the sizing of the thrusters which need to produce a radial force to maintain the circular orbit. This requirement should be considered a starting point and will need to be revised when the operational details of the manned spacecraft are better developed. Requirement 9 is quite general and expresses an ideal. The design has the crew controlling the manipulator arm operations. It is desirable to minimize the degree of interaction required. For example, it is preferable 2 of 31 American Institute of Aeronautics and Astronautics

Table 1. MaintenanceBot requirements

Level 1 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Requirement Provide autonomous visual surveillance of the exterior of the manned spacecraft Remove and replace external components under astronaut control Operate autonomously when the manned spacecraft is coasting Recharge batteries and refuel from the manned spacecraft Autonomously dock with the manned spacecraft Autonomously maneuver in close proximity to the manned spacecraft Maintain formation with any number of MaintenanceBots Orbit manned spacecraft once per hour at a radius of 50 m for up to 1 day Provide a seamless link between autonomous and crew operation of the MaintenanceBot Be repairable by the crew

to have the crew command to grasp the bolt rather than actually move the arm using joysticks or by means of sensors attached to the crew members arm. Requirement 10 requires that it be possible to bring the MaintenanceBot onboard the manned spacecraft and repair it by replacing components, fixing the wiring, etc. This rules out the use of any dangerous fuels, such as hydrazine and that it be small enough to fit inside the habitable volume while repair work is underway.

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 blocks is used to determine the relative state and attitude. Relative state is with respect to all spacecraft in the vicinity while relative attitude is with respect to the manned spacecraft. The crew interface is also carried via the ISL. The maneuver planner can command formation flying or close maneuvers depending on the particular tasks 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 includes docking and separation. The crew also drives the arm control and the attitude control system. Although shown as separate blocks since attitude control can be performed with the arm fixed, the system is actually performing 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 with one of the arms. The momentum, arm and attitude control drive the SCMGs through a torque distribution algorithm. A force 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-1 spacecraft.? 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. A.

Low-Power, Reactionless Attitude and Joint Control with CMGs

Theoretical Development

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


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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 rotor on 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 Space Station. 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 come single-gimbal CMGs will provide attitude control for several commercial earth-imaging satellites, such as Lockheed-Martin’s Ikonos and Ball Aerospace’s WorldView spacecraft. We propose to base the design of the 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 reaction wheel (RWA). An RWA applies torque simply by changing its rotor spin speed ωs , but in doing so imparts shaft power Ps Ps = τ · ωs (2) a result that assumes an entirely mechanical, lossless system. The scalar Ps is the projection of the vector motor torque τ onto the rotor angular-velocity vector ω. In contrast, a CMGs gimbal motor is roughly orthogonal to the rotor spin axis, and the resulting shaft power is virtually zero if the gimbal inertia and the motor losses are negligible. For a few hundred watts and about 100 kg of mass, large CMGs have produced thousands of Nm of torque, enough to flip over an SUV.4 A reaction wheel of similar capability would require megawatts of power to produce torque at speed. Operating either a CMG or an RWA produces a torque that reacts onto the spacecraft body, influencing the spacecraft angular momentum. The difference is that the CMG’s own angular momentum changes in 4 of 31 American Institute of Aeronautics and Astronautics

Figure 3. Honeywell M50 CMG

direction (but not in magnitude), while the RWA’s angular momentum changes in magnitude (but not in direction). If the CMG’s gimbal is rigid, the gyroscopic torque for which a CMG is responsible is purely a constraint torque. As such, it does no work. At the heart of this surprising result is the counterintuitive fact that one can alter the distribution of momentum among bodies in a dynamical system in a way that is independent of energy. That is, changing the angular momentum of various links in the MaintenanceBot system can 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 reactionless motion of high-agility gimbals and entire spacecraft payloads. The principle of operation is simple: rather than using a motor that reacts the drive torque of a moving component back onto the spacecraft bus, where it must be dealt with as an attitude disturbance, a reactionless drive absorbs the momentum internally. For example, a gimbal may be actuated by an RWA (realized, perhaps, as a circumferential ring) aligned with the gimbal axis. When the RWA spins up in one direction, the gimbal spins up in the other. The concept is shown 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 spacecraft payload: a collection of RWAs manages the entire payloads angular momentum state in three degrees of freedom, so that the payload may undergo attitude motions that are largely imperceptible to the rest of the spacecraft. Such an architecture simplifies design and integration because payload components may be developed independently, without the risk of unwanted interactions after the system is built. It also simplifies operations. Many tasks can be undertaken simultaneously with virtually no coupling between physical behaviors or tasking. In the case of the MaintenanceBot, these tasks may include simultaneously manipulating many different components of a spacecraft under repair. Reactionless benefits come with the CMG-driven MaintenanceBot. Here, incorporating CMGs throughout the kinematic chain of links provides not only high torque actuation but also inherent reactionless dynamics. When a joint is actuated, the torque comes not from a direct-drive motor that interacts with its neighboring body but from manipulating the distribution of momentum among the CMGs and the body to which they are mounted. Perhaps the most important impact of reactionless CMG-based control is that it requires only 1%-10% the electrical power for a comparable RWA-based or joint-driven robotic system, as


we explain in this section. This feature enables high agility (or dexterity) for modest power or typical agility for considerable power savings over existing robotic concepts. Low power improves the robustness and safety of 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 CMGs for each rotational joint in the MaintenanceBot. A scissored pair is an array of two CMGs with parallel gimbal axes and opposite angular velocities. Equivalently, a scissored pair may be said to consist of two CMGs with antiparallel gimbal axes and equal angular velocities. Figure 5 is a sketch of the concept. The scissored-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, the relative angle between the CMG gimbal angles φ1 and φ2 are kept constant, either through mechanical means such 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 momentum exchange from the CMG to the jointed body accelerates the body about its joint axis only, without coupling into 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 is shown in Figure 6, where the end-effector is represented as a black cone (suggesting, perhaps, the sun shield for 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 single mathematical degree of freedom for singularity avoidance, providing a singularity-free region of operation within 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 angularvelocity vector does not cause gyroscopic torques along the CMG gimbal axes, which would introduce large holding-torque requirements on the gimbal motors. Conversely, reactionless control of outboard joints virtually eliminates control power required on inboard joints (and the base body). For example, consider a


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 vector and to the CMG gimbal axis. In the case of the single body in Figure 6, one such axis is the joint axis when the CMG has rotated about its gimbal roughly 90 deg from the orientation shown. Such rotation applies torque back onto the CMG gimbal motor, which the motor then must react. Unless the motor has some mechanical antibacklash device (like a ratchet), the motor must apply this torque through electrical power in 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 attitude control is coupled to the joint attitude control to minimize power (or keep power within constraints for degraded 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 the motion 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-pair configuration is subject to the same constraints, but the example is clearer for single CMGs, and therefore we proceed with the simpler case. Let ω i/N represent the ith bodys angular-velocity vector in an inertial frame N . Let gî represent the ith CMGs gimbal axis, and let hri represent the ith CMGs angular-momentum vector. The angle φi is the ith CMGs gimbal axis, taken to be zero when hi k a î , 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 total angular velocity of the ith body is then the sum of the joint rates along the kinematic chain back to the base body, whose inertial angular velocity is ω B/N ω B/N = ω i/i−1 + ω i−1/i−2 + ... + ω 2/1 + ω 1/B + ω B/N

(3)

We assume that the CMG gimbal axes are perpendicular to the joint axes (i.e. gî ⊥ a î ). Expressed algebraically, the requirement for minimum power is then (4) ω i/n × hri · gî = 0 i.e., the projection of the gyroscopic torque due to the inertial rate of the joint onto the CMG gimbal axis is zero. Writing these constraints for all of the joints in the system results in n equations in the n joint angular


rates θ˙      

khrn k sin φn khrn−1 k sin φn−1 .. .



  =  

gˆn · (hrn × an−1 ) gˆn · (hrn × an−2 ) ... gˆn · (hrn × a1 ) 0 gˆn−1 · (hrn−1 × an−2 ) . . . gˆn−1 · (hrn−1 × a1 ) .. .. .. .. . . . . khr1 k sin φ1 0 0 0 gˆ2 · (hr2 × a1 ) −ˆ gn · hrn × ω B/N  −ˆ gn−1 · hrn−1 × ω B/N   ..   . B/N −ˆ g1 · hr1 × ω



θ˙n

 ˙   θn−1  .  .  . θ˙1

     

(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 the forward kinematics. We also note here that if the rotor angular-momentum is the same from one CMG to the 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 joint rate. 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:    gˆn · (hrn × an−1 ) gˆn · (hrn × an−2 ) ... gˆn · (hrn × a1 ) θ˙n    0 gˆn−1 · (hrn−1 × an−2 ) . . . gˆn−1 · (hrn−1 × a1 )   θ˙n−1       .. .. .. ..    ..  = . .    . . .  ˙ 0 0 . . .  gˆ2 · (hr2 × a1 ) θ1  (6) B/N ˙ khrn kθn sin φn − gˆn · hrn × ω    khrn−1 kθ˙n−1 sin φn−1 − gˆn−1 · hrn−1 × ω B/N    ..     . khr1 kθ˙1 sin φ1 − gˆ1 · hr1 × ω B/N

where

gˆ1 · hr1 × ω B/N = khr1 kθ˙1 sin φ1

(7) B/N

Consider the case of sin φ1 = 0 Here, the inboardmost joint rate θ˙1 is undefined unless gˆ1 ⊥ hr1 × ω or hr1 k ω B/N , in which case θ˙1 is merely unconstrained and can therefore take on any value. This principle can be extended for all of the joint angles, although simple expressions are not available (they involve the singular 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, what is the best strategy for accommodating the minimum-power requirements of all of them simultaneously? One approach is to constrain either the joint rate θ˙1 or the gimbal angle φ1 of each arm so that the fifth equation is satisfied. However, doing so essentially prevents the innermost joint from moving to place the end effector in a desired trajectory; it also introduces a constraint that cascades throughout the rest of the bodies, constraining their motion as well. The solution must weigh the power-minimization needs of all joints and the central body simultaneously with the combined attitude-control and joint-control commands to be executed. Absent the nth constraint, the system has a single-degree-of-freedom null space within which maneuvers can take place with minimum energy. Since far more degrees of freedom are necessary for useful tasks, the control-design problem is to find the path that minimizes energy subject to a weighted combination of 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 angular velocity ω B/N so that it minimizes the power for arbitrary joint velocities. This angular-velocity vector is regulated by the attitude-control system for the MaintenanceBot, and the operations concept may allow the use of the central body for energy-minimization. We emphasize, however, that this approach is efficient only


if the power required to steer the central body in this fashion is less than what such steering saves in the joints. To develop this law we coordinatize the vectors such that the projection of some arbitrary vector v onto each of a set of basis vectors C is written as the 3 × 1 matrixC v, i.e., h i C v = c 1 · v c2 · v c3 · v (8)

For example, these basis vectors may conveniently describe the orientation of the central body relative to an object in the workspace to be manipulated. In any case, the central body angular velocity in C coordinates C B/N ω can be specified in terms of arbitrary joint angular rates in a way that minimizes power as follows:   −C gˆnT C hrn   C TC  − gˆn−1 hrn−1  C B/N   ω = ..    .      

     

−C gˆ1T C hr1 C khrn k sin φn gˆn · khrn−1 k sin φn−1 .. .

khr1k sin φ1 θ˙n  θ˙n−1  ..   .  θ˙1

C

hrn ×C an−1 0 .. . 0

C

gˆn · C gˆn ·

C

... hrn ×C an−2 C C ... hrn × an−2 .. .. . . ... 0

C

hrn ×C a1 C hrn−1 ×C a1 .. . C C gˆ2 · hr2 ×C a1

gˆn · C gˆn−1 ·

C

     

(9) Where the superscript × indicates the skew-symmetric matrix equivalent of the cross product operation, and the superscript + denotes the Moore-Penrose pseudoinverse. The angular-velocity vectors representation in C includes only three parameters; therefore, a robot with more than three links cannot experience minimumenergy dynamics. Instead, the pseudoinverse used in this expression provides a least-squares best C ω B/N given the possibly conflicting constraints. Again, the controls architecture may instead choose to weight this constraint 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 singularities associated with robotic systems, whereby certain joints align in a way that would demand unrealizable actuator forces. Instead, when the power mapping discussed here becomes singular, certain joints simply cannot 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 of redundant joint degrees of freedom and singularity-avoidance apply here. The benefit of the MaintenanceBots architecture 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 MaintenanceBot to 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 of outboard joints lies on the axis of the inboard joint, with the result that forces and mass-center motions are irrelevant. This principle is a worthy design goal, albeit difficult to achieve in practice (particularly for times when the manipulator is carrying a payload). Nevertheless, we argue that this simplification helps make the power comparison clearer for the sake of illustrating the concepts capabilities and is therefore justified in the context of this paper. For the same reason, products of inertia are taken to be zero. Other parameters in the example are listed in Table 2. The design on which these parameters are based is somewhat arbitrary; but it 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 bus remain inertially fixed in each case: i.e., any reaction torques from the robotic arms must be taken out by the base-body attitude control system (ACS). 9 of 31 American Institute of Aeronautics and Astronautics

Table 2. Parameters for the Two-Link Example

Parameter Value Central Body Inertia Link 1 inertia Link 2 inertia Link 3 inertia CMG rotor momentum Link 1 CMG gimbal axes Link 2 CMG gimbal axes Link 3 CMG gimbal axes RWA rotor inertia CMG gimbal inertia CMG rotor rate (constant) Reference Configuration Link 1 joint axis Link 2 joint axis Link 2 joint axis End-effector location Initial CMG pair 1 gimbal angle Initial CMG pair 2 gimbal angle Initial CMG pair 3 gimbal angle

50 kg/m2 in all axes 10 kg/m2 in all axes 10 kg/m2 in all axes 10 kg/m2 in all axes 50 Nms (Honeywell M50 CMG) [0 1 0] in Link 1 coordinates [0 0 1] in Link 2 coordinates [1 0 0] in Link 3 coordinates 0.08 kg/m2 in all axes 0.02 kg/m2 in all axes 6000 RPM [1 0 0] in central-body coordinates [0 1 0] in central-body coordinates [0 0 1] in central-body coordinates [0 1 0] in central-body coordinates 0 (momentum aligned with Link 1 0 (momentum aligned with Link 2 0 (momentum aligned with Link 3

relative to Link 2 location joint axis) joint axis) 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 50 Nms). It turns out that the base body does not move (regardless of the motion of the arm), so the base-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 reaction wheel (RWA). Once again, the base-body ACS is irrelevant. 3. The third case includes traditional direct- or geared-drive motors that actuate the joints. Reaction torques applied the base body are significant, and they are compensated by a high-bandwidth reactionwheel based h ACS.√The i ACS uses four RWAs that are identical to those on the joints and whose spin 1 1 2 axes are ± 2 ± 2 2

The joints kinematics are varied numerically across a wide range in an effort to capture the worst-case power and nominal statistics. This variation forms a Monte Carlo analysis, where the joint angles, angular rate, angular acceleration, and angular jerk are given values within the limits shown in Table 3. We take these agility requirements to specify the joint kinematics, not the inertial kinematics. For example, the angular-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 point is that the CMG gimbal-motor control-loop bandwidth is taken to be much higher than the characteristic frequencies in the kinematics (e.g. the 2 rad/sec3 jerk), which is a reasonable assumption. This point allows us 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 like a day in the life of the MaintenanceBot, a statistically representative selection of maneuvers; and a simple maximum 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, some additional multiplier (say 50%) should be added to account for various IR2 losses in harness and friction losses in bearings. This scale may not be quite the same for all systems but, it turns out, the savings are so great for the baseline architecture that the difference could not change the outcome of a trade study. 10 of 31 American Institute of Aeronautics and Astronautics

Table 3. Agility Requirements for the MaintenanceBot Monte Carlo Example

Parameter Joint angle (3 independent values) Joint rate (3 independent values) Joint acceleration (3 independent values) Joint jerk (3 independent values)

Minimum Value -2 rad/sec -2 rad/sec2 -2 rad/sec3

Maximum Value 2 rad/sec 2 rad/sec2 rad/sec3

One rarely encounters a specification of jerk in this context. For the MaintenanceBot, there are several important 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. Minimizing jerk reduces the systems engineering effort in managing structural stiffnesses in the design and considering frequency-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, the maximum positive joint rate and the maximum positive joint acceleration cannot be applied simultaneously if there is a jerk limit because the acceleration must change over a finite time; and the rate limit would be exceeded during the period of negative jerk. Finally, the jerk is also directly related to the CMG gimbal acceleration. So, we have selected a jerk limit that seems to balance the demands of structural dynamics with the desire for high-speed joint kinematics. We emphasize that it is the methodology that is of greatest interest here; the specific values will be refined when MaintenanceBot parameters are allocated from higherlevel performance requirements. In all cases we assume that there is no regenerative power transfer; that is, the power bus must be designed 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 extracted from 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 significant modifications to the Honeywell designs. In case 1, the scissored-pair kinematics ( φ and its derivatives) are found from momentum conservation for the kinematic chain, using the randomly selected kinematics from the Monte Carlo simulation: 3 3/N

I3 · ω

2 2/N

I2 · ω

1 1/N

I1 · ω

+ω 3/N × I3 · ω 3/N = τ3 +ω 2/N × I2 · ω 2/N = τ2 − τ3

(10)

+ω 1/N × I1 · ω 1/N = τ1 − τ2

B B/N

IB · ω

+ω B/N × IB · ω B/N = −τ1

where B indicates parameters for the base body. However, with a non-moving base body (per our assumptions), we have 1 1/N

+ω 1/N × I1 · ω 1/N = −τ2 I1 · ω 0 = −τ1

(11) N

The jerk for a given body indicates how these torque changes in time ( τ 3 ); e.g. for the third link, 333/N

I3 · ω

N +2ω 3/N × I3 · ω 3/N + ω 3/N × ω 3/N × I3 · ω 3/N = τ 3

(12)

The momentum in the ith scissored pair is

hri = 2hr sin φ1 a î Solving for momentum along the ai axis yields φi = sin−1

hri 2hr .

h˙ ri = 2hr cos φφ˙ i → φ˙ i =

(13) The torque it applies along the ai axis is

h˙ ri 2hr cos φi


(14)

and the time-varying torque (due to jerk) is ¨ ri = 2hr cos φφï − sin φi φ˙ i → φï = h

¨ ri h + tan φi φ˙ i 2hr cos φi

(15)

From these parameters, the power required by the ith CMG is 1 1/N i/N i/N ˙ ¨ ˙ ˙ + · Ir · φi gî + ω Pi = Ωr rî + φi gî + ω −Ωr rî × φi gî − Ωr rî + φi gî × ω 1 1/N −φ˙ i gî × ω i/N φ˙ i gî + ω i/N · Ig · φï gî + ω

(16)

where Ωr is the CMG rotor rate, rî is the rotor spin axis, φ˙ i is the gimbal rate, ω i/N is the ith links angular i/N

i is the velocity in an inertial frame N , Ir is the rotor inertia dyadic, φï is the gimbal acceleration, ω angular 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 the angular momentum for each link, we simply compute

hri = a î · Ii · ω i/N

(17)

i i/N i/N i/N τri = a î · Ii · ω +ω × Ii · ω

(18)

Similarly, the RWA torque is simply

The power for each RWA is then Pi = τri

hri Ir

(19)

where we have taken advantage of the fact that the rotor inertia is the same in all axes (by the assumptions explained above). In the third case, we compute the time rate of change in the three-link systems kinetic energy Ejoints and take that power to be required of the joint motors, whatever they may be. Then, the net momentum of the outboard 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 = E˙ joints + E˙ ACS =

3 X

i i/N

ω i/N · Ii · ω

+E˙ ACS

(20)

i=1

The power due to ACS activity is then simply the shaft power for each of the four RWAs. Given the matrix A of spin axes (as defined in Table 1),   0.5 −0.5 0.5 −0.5   A =  0.5 0.5 −0.5 −0.5  (21) √ √ √ √ 2 2

2 2

2 2

2 2

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



 −1  hbase  = AT AAT 

(22)

The same result applies to reaction-wheel torques τacs,i Some straightforward algebra leads to 1 2 | AAT |hbase | E˙ ACS = |τbase Ir 12 of 31 American Institute of Aeronautics and Astronautics

(23)

where the absolute values (not the vector norms) are taken for each element because of our requirement that there is no flywheel-style energy storage, and that all power must come from the base body power bus or be shunted into a resistor. The results of the Monte Carlo analysis for day-in-the-life statistics are shown in Figure 8. The first case, the baseline MaintenanceBot, requires about 151 W for typical operations, despite its extraordinarily high agility. In contrast, the reactionless RWA case requires about 3300 W, and the traditional joint-drive architecture requires about 7200W.

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

The worst-case analysis (maximum rate, acceleration, and jerk, for all values of joint angle) reveals a similar trend. The bar graph in Figure 9 shows that the CMG-based reactionless architecture requires between 200 W and 1500 W for worst-case kinematics over the range of joint angles, while the RWA-based architecture 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 outperform other systems in power for high-agility maneuvers. Furthermore, this architecture opens up a large trade space for power vs. agility, allowing highly effective MaintenanceBots to be incorporated for relatively low power-specific mass. Or, for a given mass, the MaintenanceBot can withstand more demanding operations for longer than competing architectures.

V.

CMG Implementation Issues

A CMGs output torque is proportional to its rotor angular momentum and its gimbal rate. In this application, the joint rate capability depends on this stored momentum, and the joint acceleration depends on both the CMG’s momentum and its gimbal-rate capability. High-speed rotors would therefore seem to solve many problems: for a given momentum, the rotor would weigh less; or for a given rotor mass, the momentum would be greater. Figure 10 shows the influence of rotor size on momentum-storage capability and mass efficiency (momentum per kg). The strawman rotor design considered here consists of a 1 cm deep x 3 cm high rim and a 0.5 cm thick web. This calculation does not account for the additional mass of structure, bearings, motors, and electronics associated with larger rotors. Energy storage5 applications have investigated high-speed rotors for some time, although not for gimbaled applications, where bearing loads are an issue. In the case of energy storage, material stress limits typically constrain the rotor size. In space applications, however, design robustness demands careful attention to bearing life. In fact, it is bearing issues that have been partly responsible for the International Space Station


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 is based on robust bearing designs for rotor rates from 4500 to 6500 RPM.7 Although high-speed bearing technology will doubtless continue to advance for large rotors, the MaintenanceBot is designed for high TRL and 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 devices stand some chance of providing adequate momentum for compact robotic systems. However, bearing issues would have to be addressed here, too, because directional gyros typically are not meant to transfer load across the spin bearings.8 This approach was taken in the Manned Maneuvering Unit (MMU), which incorporated three scissored pairs of CMGs developed by David Osterberg of Honeywell Defense and Space Electronics Systems (Sperry at that time). Figure 11 shows a layout in the MMU. The three oblong boxes contain the modified directional gyroscopes.

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


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 manned spacecraft The model in case 1 is for a rigid body with the CMGs. Considering a single CMG, the equation for the momentum is H = A(Iω + Bhcmg ) (24) where hcmg is the vector momentum of the CMG in the CMG frame. B is a transformation matrix that transforms from the CMG frame to the core body frame. A transforms from the body frame to the inertial frame. I is the total vehicle inertia and is assumed constant, even with the CMG rotation. The dynamical equations are then ˙ cmg hcmg + ω × (Iω + Bucmg hcmg ) T = I ω˙ + Bu (25) where hcmg is assumed constant ucmg is the unit CMG vector and T is the external torque due to thruster firings. Interestingly, if all of the CMGs have the same momentum vector we can generalize this equation to any 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× where



0  Ω× =  Ωx Ωy

−Ωx 0 Ωz

(26)  Ωy  −Ωz  0

(27)

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


the CMG momentum is around z. Therefore the derivative term becomes   0 ˙ ˙ cmg hcmg = BF  Bu  − cos θ  θh cmg − sin θ and the momentum term becomes

 0   B˙ = BF  − sin θ  hcmg cos θ 

VII. A.

(28)

(29)

Relative Sensing

Introduction

The MaintenanceBot uses two systems for relative attitude and position sensing. The primary system is an RF system that measures range and range rate from the MaintenanceBot to the manned spacecraft and computes relative attitude and position from those measurements. The manned spacecraft has 8 transmitters arranged on booms for this purpose. The RF system uses the same antennas and receivers that are used for RF 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 spacecraft and compares it against an image generated onboard the MaintenanceBot from a computer model of the manned spacecraft. This allows fine pointing and position control near the manned spacecraft. Since the MaintenanceBot always knows its position and attitude from the RF system the vision system need only make 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.

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Figure 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 manned spacecraft. To achieve sufficient position dilution of precision (PDOP), generally taken to be less than 6, the RF system must have a sufficient antenna baseline to cover desired orbital distances and enough antennas to assure a solution at any location. The geometric dilution of precision for pseudorange measurements is defined from the linearized navigation solution from at least 4 pseudorange residuals. q GDOP = trace[H T H]−1 (30)

where the rows of H are defined using the directional cosines from the MaintenanceBot to each antenna, hi = [ u î

−1 ]


(31)

The position dilution is computed by summing only the position elements in the trace operation, excluding the time dilution. Consider a system of six antennas, arranged in two planes, each plane an equilateral triangle, as shown in Figure 13.

z

50

0 100 50 0 −50 80

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−50 40

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−100

−40

x

y

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 larger than the antenna baseline. 25

PDOP with Orbital Radius for Antenna Baseline of 100 m

30

PDOP Variation with Baseline and Distance 50 100 150

25

20 50 100 200

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PDOP

PDOP

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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, azimuth and elevation from a ladar. These are input to an iterated extended continuous discrete Kalman Filter which estimates both the relative state and the absolute state of the manned vehicle. The manned vehicle orbit is determined from range and range rate measurements from ground stations (using standard multiple tone ranging) or from navigation satellites. Currently navigation satellites are only available around the earth but it is expected that navigation satellites will be placed in orbit around the Moon and Mars to support future


missions. When the two vehicles are closer the CAD based algorithm returns relative position supplemented by range measurements until the minimum range is reached. The relative attitude determination system incorporates manned vehicle gyro measurements and any external 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 the general concept of optimal image matching demonstrated to within the capability of MATLAB’s image software and the optimization method used.

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Figure 15. Vision System

An onboard CAD model generates an internal representation of the manned vehicle. Centroiding is applied to this model and the image in the camera to get a rough position estimate used to start the iterative process. A range measurement is used if available. The CAD and camera image are filtered. Filtering is designed to enhance the image to speed convergence of the optimization algorithm. Filtering may include a filtering (e.g. Gaussian), edge detection and point detection algorithms. The filtered images (which may include color images, edges or points) are processed to get the attitude and position. If sufficient points have been identified a point algorithm can be used to determine the position and orientation directly. The resulting position and orientation is fed to a Kalman filter which maintains the state of both the MaintenanceBot and manned 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, and it provides a constant circumnavigation radius. The dynamical equations for planar motion, written in cylindrical coordinates are r¨ − rθ˙2 = ar

(32)

rθ¨ + 2r˙ θ˙ = 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 acceleration. The desired accelerations are ar = −rθ˙2 − k¨ r (36) 18 of 31 American Institute of Aeronautics and Astronautics

and

aθ = rθ¨c

(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 q a = r a2r + a2θ (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 to a slight offset since the controller does not feedback the radius. Small errors in radius are acceptable. The radial rate is damped to zero very quickly. The circumferential angular velocity reaches a constant and the angle 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. It then drops to the magnitude required to maintain the radial position. The pointing angle decreases slowly then jumps to -90 degrees when the desired circumferential angular velocity is achieved. State

Control

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Figure 16. Close maneuver acquisition simulation

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

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Figure 17. Radial Force


The A* search algorithm can be used to deviate from a constant radius to avoid stayout zones. This would be necessary during an inspection of the manned spacecraft since the MaintenanceBot would want to stay out of the sensor fields-of-view, RF transmitter booms, etc. A* searches in a state space, in this case a grid around the target at the desired radius, for the least costly path from a start state to a goal state. The cost during a maneuver is the change in angle needed to go from one state to another. A* is guaranteed to 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 the heuristic function so that no search that uses the same function and finds optimal paths will expand fewer nodes than A*.11 The algorithm requires two lists of states called open and closed for unexamined and examined states. At the start closed is empty and open has only the starting state. In each iteration the algorithm removes the most promising state from open for examination. If the state is not a goal then the neighboring locations are sorted. If they are new they are placed in open. If they are already in open the information about the states is updated if this is a cheaper path to those states. States that are already in closed are ignored. If open becomes empty before the goal state is reached then there is no solution. The most promising state in open is the location with the lowest cost path through that location. This heuristic search ranks each node by an estimate of the best route that goes through that node. The typical formula is 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 start and h(n) is the heuristic estimate of the cost to the goal from n In this problem the state space is a set of points 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 locations on the sphere. Figure 18 shows the path. Each color is a segment of the path. As can be seen the selection of grid points is critical as it determines the actual path. Denser grids require more computation but would result in smoother paths. The ideal grid would have more points around obstacles and fewer points in open areas. There is also nothing the prevent the use of 3-dimensional grids if it is desired to change the radius of the orbit around the target vehicle. A* Path

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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 difference between the measured relative state and the output of the filter. This serves to smooth the inputs to the controller 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 20 of 31 American Institute of Aeronautics and Astronautics

(43)

and the propagation of the controller state is xk+1 = axk + bu

(44)

f = −my

(45)

The commanded force is More precise tracking is possible using a PID (proportional integral differential) controller but would only be 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% higher than a maneuver without a stayout zone. If there are no stayout zones the A* maneuver requires the same delta-v as an ideal circular maneuver with constant radial thrusting. This is despite the fact that the cost does not explicitly involve velocity change. Tracking performance is shown on the right-hand side of figure 19. The controller smoothes out the abrupt changes in the trajectory requested by A*. The abrupt changes are due largely to the grid. A PID or higher gain controller would track the trajectory more precisely at the cost of more fuel consumption. Delta−V

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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 multiple MaintenanceBot˜concept. In general, formation flying involves actively controlling the relative position and velocity of two or more spacecraft to establish a desired geometry. Examples include space-based interferometery and synthetic aperture radar, each of which utilize large separation distances between spacecraft to conduct high-resolution sensing. For the majority of the time on-orbit, each MaintenanceBotwill ˜ fly in formation with the manned spacecraft. 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 MaintenanceBots˜in formation is so that they may provide continuous visual inspection. An additional motivation is to enable a more rapid response to emerging events that require immediate inspection and/or service. In order to minimize fuel-usage, the MaintenanceBotswill ˜ follow natural repeating trajectories. These are closed trajectories relative to the prime spacecraft that repeat each orbit period and, in the absence of disturbances, require no applied force to maintain. Relative motion is defined within the Hill’s coordinate frame – a non-inertial, rotating frame where the origin is fixed to the manned vehicle. Figure 20 shows the relative frame, hereafter referred to as Hill’s frame. The x-axis points in the zenith direction, the z-axis is aligned with the angular momentum vector, normal to the orbital plane, and the y-axis completes the right-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 relative trajectories can be readily found. The constraint for achieving periodic relative motion with two satellites is 21 of 31 American Institute of Aeronautics and Astronautics

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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 argument of perigee, relative motion in the orbit plane can be achieved. In circular orbits, this in-plane motion is sometimes 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 right ascension and/or inclination. The geometry of the relative motion in a circular is easily expressed as a superposition of along-track offset, in-plane elliptical motion, and cross-track oscillation. The following five parameters are used to fully define the geometry of any type of relative trajectory with a circular reference orbit: Table 4. Geometric Parameters for Relative Motion in Circular Orbits

Parameter y0 aE β0 zi zΩ

Description Along-track offset. Defines the center of the in-plane relative ellipse. Semi-major axis of relative ellipse. Phase angle on relative ellipse at ascending equator crossing. Measured positively from −x axis to +y axis of Hill’s frame. Cross-track amplitude due to inclination difference. 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 American Institute of Aeronautics and Astronautics

orbits. The in-plane motion no longer follows a 2 × 1 ellipse, and the cross-track oscillation is not necessarily centered about the origin. The shape of the trajectory depends upon the eccentricity, as well as the points in the orbit where the maximum radial and cross-track amplitudes occur. The geometry of the relative trajectory can be defined using the parameters in Table 5. Table 5. Geometric Parameters for Relative Motion in Eccentric Orbits

Parameter y0 x ¯ νx¯ z¯ νz¯

Description Center of along-track motion Maximum radial amplitude True anomaly where x ¯ occurs Maximum cross-track amplitude True anomaly where z¯ occurs

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

_

z 2 z [km] 1 Cross− Track 0 _

−1

x −2

−2 −2

−1 −1

y

0 y [km] Along−Track

0

0

1

1

2

2 3

x [km] Radial

3

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-track motion 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 an eccentric anomaly of E = 90 deg). This trajectory is shown in Figure 23 on the following page. Note the considerable difference in the motion when projected onto x-z and x-y planes. The symmetric figure-eight shape in the x-z plane is achieved by forcing z¯ to occur at E = 90 deg. For a given number of MaintenanceBots˜flying in formation with the manned vehicle, we wish to design a set of relative trajectories that meet the following objectives: 1. Maintain a minimum “safe” separation distance between the manned spacecraft and all other MaintenanceBots 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 the parameter set for each trajectory. 23 of 31 American Institute of Aeronautics and Astronautics

_

z 2 z [km] 1 Cross− 0 Track −1 _ −2

x

−3 −4 −2

y 0

0

y [km] Along−Track

R[htb]

−2

0

2 2

4

x [km] Radial

Figure 23. Example Relative Trajectory in an Eccentric Orbit

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

0.1 0.05 z [km] 0 −0.05 −0.1 0.2 0.1 0 0.1

−0.1 0 y [km]

−0.2

−0.1 x [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, as well as the number of MaintenanceBots. The fourth item, however, concerns the nature of relative orbit dynamics, and depends upon the orbit. The most significant relative disturbances that will impact formation flying 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. The induced 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 be much greater than that of the MaintenanceBots. It is possible, however, to intentionally design a relative trajectory with inclination difference so that the drift from J2 serves to partially counteract the differential drag. The MaintenanceBots must maintain their desired relative trajectories in the presence of these differential disturbances. Because formation flying will represent a significant portion of the time on-orbit, it is important


to devise a formation maintenance strategy that requires as little delta-v as possible. The general approach is to define an allowable position deadband, or error-box, for each MaintenanceBot, so that as long as it stays within the box it does not maneuver. Once the position error exceeds this deadband, a corrective maneuver is planned and implemented. This on-off control approach leads to a limit-cycle behavior, where the frequency depends upon the accuracy of the relative state estimate and the size and direction of the relative 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 forms in recent years.12, 13, 14, 15 In general, the LP approach is used to compute an impulsive delta-v sequence over a fixed time window, so that the desired state is reached at the final time and a given cost function is minimized. The cost function may be defined to simply represent the total delta-v for the maneuver, or it may 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 in state-space. The relative dynamics for a circular reference orbit are linear time-invariant (LTI), and may be expressed as: ˙ x(t) y(t)

= Ax(t) + Bu(t) = Cx(t)

(46)

where x is the state vector, u is the control input, and y is the output. The state vector consists of the h i⊤ 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:     0 0 0 0 0 0 1 0 0      0 0 0   0 0 0 0 1 0       0 0 0   0 0 0 0 0 1      (47) B= A=  2 0 0 2n 0    1 0 0   3n 0     0 0 −2n 0 0   0 1 0   0 2 0 0 1 0 0 −n 0 0 0

where n is the mean orbit rate. The continuous-time system is discretized using a zero-order hold over a time-step ∆t to obtain: xk+1 yk

= Axk + Buk

(48)

= xk

where A and B now denote the discrete-time state-space matrices. The expression for the second state can be found as follows: = Ax1 + Bu1

x2

= A (Ax0 + Bu0 ) + Bu1 = A2 x0 + ABu0 + Bu1

(49)

Extending to the N th state, we have:

xN = AN x0 +

h

N −1

A

B

N −2

A

B

· · · AB

i



  ×  

For future convenience, let us introduce the following definitions: h i ´ = B AN −1 B AN −2 B · · · AB h i⊤ u ´ = u0 u1 · · · uN −1 25 of 31

American Institute of Aeronautics and Astronautics

u0 u1 .. . uN −1

     

(50)

(51)

so that

ú xN = AN x0 + B ´

(52)

th

This gives us an expression for the N 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 the above expression as two inequalities. xN − x∗ xN − x∗

≥ ≤

−ǫ +ǫ

(54)

ú AN x0 + B ´ − x∗ ≥ −ǫ N ú A x0 + B ´ − x∗ ≤ +ǫ

(55)

ú −B ´ ´ Bu ´

(56)

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

Further algebraic manipulation yields: ≤ ǫ + AN x0 − x∗ ≤ ǫ − AN x0 − x∗

Now let A˜ and ˜ b be defined as: #

A˜ =

"

´ −B ´ B

˜ b =

"

ǫ + AN x0 − x∗ ǫ − AN x0 + x∗

(57) #

so that the inequality may be written as follows: ˜u ≤ ˜ A´ b

(58)

The problem is now posed in a form suitable for the Simplex algorithm, which is a well-known technique for solving LP problems. The objective is to minimize the cost c˜u ´ subject to the constraint defined in Eq. (58). Here, the cost coefficient vector, c˜, has 3N columns, and is nominally composed of all 1’s. The weights may be adjusted to provide a greater or lesser penalty at different times, or for different axes in the relative frame. The approach for setting up the LP problem for eccentric orbits is described in another paper.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 a nuclear thermal vehicle with hydrogen fuel using the Small Engine Reactor developed by Los Alamos.17 The vehicle is designed to put a 125,000 kg payload into low Mars orbit. It is assumed that the vehicle is refueled at Mars. Unlike nuclear electric vehicles, nearly all of the reactor power goes into the fuel so relatively little radiator area is needed. The vehicle carries 6 crew members to Mars orbit where they descend to the surface in the lander which uses direct thrust for landing. The lander would shuttle fuel back to the transfer vehicle during 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 quarters are a ring which spins to provide some artificial gravity. The Small Engine Reactor is seen on the left hand side. The engine has a 2π steradian radiation shield so it must be approached from the nozzle end by the MaintenanceBot. The hydrogen fuel provides shielding, of varying thickness, during most of the mission. The delta-V budget is based on impulsive burns. This approximation is justified since the engine can consume all of the fuel in less than a day. 26 of 31 American Institute of Aeronautics and Astronautics

Table 6. Hypothetical Mars Transfer Vehicle (MTV)

Property Transfer Vehicle DV Earth Escape DV Earth to Mars DV Mars Capture DV Total Lander DV to 15 km DV 15 km to 100 km DV Total Lander Mass Mass Dry Mass Payload Mass Fuel Mass Total Transfer Vehicle Mass Mass Dry Mass Fuel Mass Payload Mass Total Thrust Exhaust Power

Value 3.15 km/s 5.59 km/s 1.45 km/s 10.20 km/s 3.76 km/s 0.04 km/s 3.81 km/s 51,711 kg 4,000 kg 73,024 kg 129,395 kg 124,066 kg 579,839 kg 129,395 kg 833,960 kg 72 kN 308 MW

XI.

MaintenanceBot System Design

The MaintenanceBot has 24 cold gas thrusters, 4 SCMGs (3 orthogonal and 1 skew), a lithium ion battery, dual triple junction solar panels and an ITT Low Power Transceiver (LPT) for communications and GPS processing. The LPT can handle up to 12 antennas. There are 8 GPS antennas and 4 omni antennas for intervehicle links and links to the ground. The docking apparatus is conceptually similar to the system developed by Prof. Michael Swartout18 and involves 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 the manned spacecraft while it is working. This allows the MaintenanceBot to unload momentum without firing thrusters. 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 include replacement of CMGs and other components. The ability to replace CMGs reduces the lifetime requirements on the bearings thus reducing the need for very sophisticated bearing assemblies. Since work would be done on the MaintenanceBot inside the bay when it was pressurized, no special equipment would be needed for repairs, 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 MaintenanceBot to maintain a steady attitude rate around the y-axis will the z-axis thruster fire to maintain a steady radial


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 CMG and T is the torque demand. Each column of U is a function of the current CMG angle. Each CMG is represented 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 with small angles. This is not the most sophisticated CMG steering law but can be implemented with only a few lines of code. The control algorithm is a rate controller with an additional term to cancel the Euler coupling of the CMGs. 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 the MaintenanceBot pointing at the manned vehicles.

XIII.

Conclusion

A preliminary design for a maintenance robot for manned vehicles is presented in this paper. It makes use of single gimbal control moment gyros for attitude control and robot arm movement and cold gas thrusters for formation flying and close maneuvering. CMGs present a clear advantage over reaction wheels or thruster control for maneuvering of the core body. Use of the CMGs for the robot arms is feasible with existing CMGs but 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 development of algorithms (i.e. inverse kinematics) to position the arm. The various control systems described in this paper will be integrated and integrated simulations of the system will be conducted. The crew interface will be defined, developed and tested as part of the simulation. With regard to the CMGs studies will continue on developing more compact CMGs for arm actuation. 28 of 31 American Institute of Aeronautics and Astronautics

Figure 26. MaintenanceBot

Table 7. MaintenanceBot components

Component SCMGS Integrated Avionics Unit Battery Gyro Camera Thrusters Comm Temperature Transducers

Model N/A RAD 750 Lithium ion ADXRS150 1 50-673 1 40

Number 4 + 2 per manipulator dof 1 1 3 TPS 24 LPT AD590

Vendor Cornell Broad Reach/BAE Systems Japan Storage Battery Ltd. Analog Malin Moog ITT Analog Devices

Reference 19 20 21 10 22 9 23

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

References 1 NASA/JSC,

“Robonaut,” World Wide Web, http://vesuvius.jsc.nasa.gov/er er/html/robonaut/robonaut.html, 2004. M., “Dextre,” World Wide Web, http://www.space.gc.ca/asc/eng/media/press room/, 2004. 3 “Honeywell M50 Flysheet,” Tech. rep. 4 Quartararo, R., “Course Notes, Introduction to Spacecraft Attitude Control,” Tech. rep., 2003. 5 McLallin, 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. 8 Peck, M., Miller, L., and et al., “An Airbearing-Based Testbed for Momentum Control Systems and Spacecraft Line of Sight,” Feb 2003. 9 ITT, Low Power Transceiver (LPT), http://www.ittspace.com/products/lowpow.asp. 2 Robotics,


Figure 27. MaintenanceBot in orbit around the manned vehicle

1

Body Rates

x 10

0

0

ω

x

θx

-3

CMG Angles

0.5

-1

-0.5 2

-2

θy

0.015

1

0.01

ω

y

0 0

θz

0.005

-0.5

0

-1

15

0.5

10

θs

ω

z

-4

x 10

5

0 0

-0.5

0

1

2

3

4 5 Time (min)

6

7

8

-5

9

0

1

2

3

4 5 Time (min)

6

7

8

9

Figure 28. MaintenanceBot in orbit around the manned vehicle

10 Systems, M. S. S., MSSS Offers Space-Qualified Camera Systems For All Mars Scout, Discovery, and New Frontiers Mission Requirements, http://www.msss.com/camera info. 11 Stout, W. B., The Basics of A* for Path Planning, Charles River Media., 1999. 12 Tillerson, 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. 13 Breger, L., Ferguson, P., and How, J., “Distributed Control of Formation Flying Spacecraft Built on OA,” AIAA Guidance, Navigation and Control Conference and Exhibit, Austin, TX, 2003. 14 Richards, 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. 15 Tillerson, M. and How, J., “Formation Flying Control in Eccentric Orbits,” AIAA Guidance, Navigation and Control Conference and Exhibit, Montreal, Canada, 2001. 16 Mueller, J. B., “A Multiple-Team Organization for Decentralized Guidance and Control of Formation Flying Spacecraft,” No. AIAA-2004-6249, Sep. 17 Joseph A. Angelo, J. and Buden, D., Space Nuclear Power , Orbit Book Company, 1st ed., 1985. 18 Fitzpatrick, T., Satellite as small as a cantaloupe docks with mothership size of a medicine ball , http://mednews.wustl.edu/tips/page/normal/4147.html. 19 Broad-Reach, Integrated Avionics Unit, http://www.broad-reach.net/iau.html. 20 Ltd, J. S. B., Lithium ion battery, http://www.nippondenchi.co.jp/npd e/lithium/lithium.html. 21 Devices, A., iMEMS Gyroscopes, http://www.analog.com/Analog Root/sitePage/mainSectionHome.


22 Moog,

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

23 Devices,


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