Control Moment Gyros as Space-Rootics Actuators Daniel Brown Cornell University, Ithaca, ew Yor, 4850 Control moment gyros (CMGs) are an energy-efficient means of reactionless actuation. CMGs operate y gimaling a high-speed rotor to change the momentum of a ase ody. We investigate a rootic linage actuated y scissored-pair CMGs. Scissored pairs constrain the output torque from the CMGs to act along the joint axis, eliminating undesirale gyroscopic reaction torques. This wor compares the energy required to actuate a rootic linage with either CMGs or direct drive motors. We show that the CMG power is equal to the directdrive power for a large range of gimal inertias and pea gimal angles. The transverse rate does not independently affect this result. We find that the scissored pair s pea gimal rate is a ratio of ody acceleration to ody rate. The equations of motion for an n-lin root with CMGs are presented in a recursive form for easy implementation in software. The results for a one-lin root extend easily to two-lin roots with orthogonal joint axes when the pea ody rate and pea gimal angle are adjusted to account for the influence of neighoring lins. CMGs surprisingly outperform direct drive for a two-lin root with parallel joint axes when the joints rotate with opposite sign, as in reaching tass; although the reverse is true when the joints act in unison. These differences arise ecause CMGs produce ody torques with a zero-torque oundary condition at the joint, whereas direct drive produces joint torques. I. Introduction EACTIOLESS actuation of a rootic assemly on a spacecraft provides several advantages over typical R actuation methods []. A reactionless system decouples the attitude control system (ACS) from the dynamics of the rootic arm. System-level pointing performance can e improved y removing the nown disturances created y a rootic arm. The root arm may include the sensors or camera eing pointed. For such a configuration, rapid rootic movements do not impart low-frequency disturances that might excite structural virations. Another enefit of reactionless rootics is that the agility required of a specific susystem need not e applied to the satellite as a whole, reducing the pea ACS torques when used with the root. This wor explores possile power-efficiency advantages of a reactionless root without maing assumptions aout the ACS power savings. Power-efficient control moment gyros (CMGs) produce a torque directly on a ody without transmitting that torque through neighoring joint axes. We compare the energy used y a root linage actuated with direct drive motors acting on the joints to that of an identical system actuated y CMGs. We also give novel results on differences etween the two systems that arise from joint axis arrangements ecause CMGs apply torques directly to each ody, not through the joints. A spinning ody resists change in oth the magnitude and the direction of spin, i.e., the change in angular momentum is equal to the applied moments. Momentum control of a ody uses the emedded momentum of a spinning rotor to produce an internal torque that causes the rest of the ody to rotate in such a way that the system s angular momentum stays constant. A ody that uses momentum control can reorient without propellant and without changing its system net angular momentum, valuale traits in spacecraft applications. The internal momentum is either a fixed-axis, variale-speed rotor (reaction wheel) or a gimaled-axis, constant-speed rotor (CMG), or oth (variale-speed CMGs) [2]. A reaction wheel assemly (RWA) changes its rotor speed only. The spin axis is fixed to the satellite. The approximate energy cost of using an RWA of inertia I r is the change in the inetic energy of the rotor. Graduate student researcher, Mechanical and Aerospace Engineering, 38 Upson Hall, AIAA student memer.
2 2 Δ Er = Ir ( ω2 ω ) () 2 where the initial and final speeds of the rotor (ω, ω 2 ) are taen relative to the satellite reference frame. In spite of the added energy cost, reaction wheels have een used extensively due to their simplicity, reliaility, and strong flight heritage. In contrast, the CMG uses a constant-speed rotor on a gimaled axis. Its rotor s inetic energy changes only insignificantly (e.g. in response to low-speed ase motions). To first order, the change in angular momentum is given y the gimal rate crossed with the rotor momentum, resulting in a torque perpendicular to oth the gimal and the rotor axes (see Fig. ). τ = φ gˆ h (2) C CMGs also have an important flight heritage, especially for producing large torques required of large structures such as ISS or MIR [3]. Much recent attention has focused on smaller CMGs for use on small satellites [4, 5]. This paper aims to encourage further development of small CMGs y highlighting their potential application to rootics. Some energy is used for CMG driven maneuvers at least as much as the change in the satellite s inetic energy. We explore how much energy is used y a CMG with a constant speed rotor, limiting ourselves to scissored-pair, single-gimal CMGs throughout this study. Comparisons of RWAs and CMGs have previously shown the power enefits of using CMGs, oth in attitude control and specifically for rootic applications [6]. Root arms usually use electric motors at the joints. A proposed ifocal relay telescope uses CMG attitude control on one memer, and connects the other memer with a joint motor [7]. CMG-actuated rootic linages are a recent idea and have een proposed for viration and slew control of a large truss [8] and three-degrees-of-freedom control of a coelostat telescope [9]. The proposed CMG-actuated roots use scissored pairs of CMGs also referred to as V-gyros or twin-gyros [0-2]. Scissored pairs produce torque aout a single axis y using mirrored gimal angles to cancel unwanted torque on the ody. Cross coupling torques acting on the gimal motors that result from ody motion can e cancelled internally to reduce gimal torque [2-4]. The present wor considers first a single-lin root to create an intuitive foundation and provide useful equations for sizing CMGs. We compare CMGs to direct-drive actuation--the liely competing technology for rootic actuation--for a one- and two-lin rootic arm. We give an explanation for novel and counter-intuitive results for which CMGs outperform direct drive in a planar two-lin arm. II. CMG Dynamics A. Isolated CMG This section develops the equations of motion for a single CMG. Similar equations have een developed elsewhere [5], ut the following first-principles derivations clarify the assumptions made and help estalish notation efore comining two CMGs into a scissored pair. Consider a ewtonian frame denoted y, a ody-fixed frame B, and the CMG gimal-fixed frame G. The CMG s angular momentum vector aout its center of mass, h c, is the comined momentum of the gimal and the rotor: h = I ω + I ω G/ c g r g gr R/ G/B ( ) r ( R/G G/B ) G/B ( ) r = I ω + ω + I ω + ω + ω = I ω + ω + h For a constant speed rotor, the time derivative of Eq. (3) taen in the frame is r h r φ ê 2 ê 3 φg ˆ τ C φ ê Figure. CMG vectors and scalars defined. (3) 2
( ) ( ( ) ) G B G/B G/B G/B G/B = + + + + + c gr gr r h I ω ω ω ω ω ω I ω ω h (4) where the following shorthand is used to indicate a vector derivative: d ω = ω (5) dt For the analysis and simulation, the CMG inertia I gr is constant in any frame as for a spherical ody. I + I = I = I (6) g r gr gr This simplification clarifies the analytical results y eliminated various minor terms in the equation and reduces the numer of free parameters in the simulations. We justify this choice for a physical system y noting that I gr comines the gimal support structure and attached framewor and motors with the rotor. The gimal rate and acceleration may e written in terms of the gimal angle φ using an over dot to denote the time derivative of a scalar. G/B ω = φ g ˆ (7) G G/B ˆ ω = φ g (8) Comining Eqs. (6) to (8) with Eq. (4) results in a more usale form: ( ) B h = I φg ˆ + ω φg ˆ ω + φg ˆ + ω h (9) c gr r The time derivative of the angular momentum must e equal to the external torques. The torques acting on the CMG are the supplied gimal torque and the torque reacted onto the ody. We neglect friction, electromagnetic, and flexile effects, instead focusing on the dynamics of the system. The gimal torque always acts aout the gimal axis whereas the direction of torque reacted onto the ody varies. Even for a stationary ody (ω=0), the CMG torque on the ody will change ased on the gimal angle (Eq. (2)). Adapting to changing CMG output torque is among the iggest challenges of CMG-ased attitude-control system design due to singularities [6]. Singularities arise when the possile output torques cannot produce the desired torque. For the system under investigation here, the desired torque is always along the joint axis. We use a scissored pair to constrain the torque output to act only along the joint axis. B. Scissored Pair In a scissored-pair, two CMGs with parallel gimal axes maintain equal-magnitude and opposite-sign gimal angles (Fig. 2). Singularities occur only if the commanded torque exceeds the capaility of the scissored pair in magnitude or if the momentum stored in the pair is at a maximum. These saturation singularities occur in any actuator. Scissored pairs also have a simple zero-angularmomentum configuration, important for rotor spin-up and ensuring that motion of other lins does not induce unwanted gyroscopic torque. We have also shown elsewhere how the ase-rate effects can e reduced in a scissored pair through a mechanical coupling that enforces the gimal angle constraint [3]. While we do not explore the application of a single CMG for each lin, we speculate that there may e some enefit to e gained y using them in this way; we set aside these issues for future wor. The magnitude of the gimal torque for the first CMG is found y taing the dot product of ĝ with Eq. (9). To Figure 2. Scissored-pair inematics. distinguish etween the two CMGs, a suscript or 2 is used. 3
The direction of the rotor momentum is different for each CMG, even though the magnitude is constant, so that h r h r2. For the other CMG, we replace φ with φ. ( ) B I φ I ˆ g gr gr r τ = + ω g + ω h g ˆ (0) τ ( ) B I φ I ˆ g2 gr gr r2 = + ω g + ω h g ˆ () The control volumes shown in Fig. 3 show that the net applied torque, τ A, is equal to the difference etween the two gimal torques when using a mechanical coupling to enforce the mirror symmetry of the scissored pair. τ = τ τ (2) τ A g g2 ( ( )) ω h h g (3) = 2I φ + ˆ A gr r r2 τ φ ω φe (4) = 2I 2h cos A gr r ˆ The first term corresponds to rotating the gimal. The latter term accounts for the ody rate and may e referred to as the ase-rate effect. The ase-rate effect plays a significant role in energy costs of CMGs. The rotor momentum is usually large relative to the gimal inertia and accelerations, with the latter limited y the gimal motor. A simplified expression for gimal torque is: τ A ω 2h cosφe (5) r ˆ The power used y the CMGs for a maneuver is determined using gimal torque times gimal rate. P = τ φ (6) A This expression neglects losses due to friction and electromagnetic inefficiencies under the liely assumption that the gimal torque per se drives the power design in an agile application. Friction losses may mae CMGs an inefficient choice for a generally quiescent system. This study does not distinguish etween positive or negative power since oth require energy from the spacecraft power system. The sign would matter in a case where the spacecraft power system efficiently recovered this energy expenditure in a regenerative fashion, e.g. using the gimal motor as a generator. We assume that such an architecture is not in place for purposes of this study. Here, power is independent of the sign of gimal torque and gimal rate. Therefore we consider the asolute value of power in our comparisons. τ A τ g τ g2 τ A = τ g τ g2 τ g gears τ g2 Figure 3. Gimal torques for a scissored pair. III. Single Lin Root We first derive an analytical expression for power use for a single lin powered y either a scissored-pair of CMGs or direct drive and show oth use the same power. In this wor, the root attaches to a stationary ase, i.e. the satellite is sufficiently large and slow to neglect the ase dynamics. We consider specifically agile rootic applications with motions up to rad/s rates that favor a reactionless root over a fixed spacecraft. We simulate a single-lin root to extend the results of the analysis to include larger gimal inertias and a non-zero transverse rate of the ase. A. Single Lin Analysis As efore, we derive the equations associated with the general case and apply reasonale assumptions to gain intuition aout the system. The motion of a single lin may e descried in terms of the angular momentum: 4
H I ω (7) = The angular momentum derivative is ( ) B H = I ω + ω I ω (8) The direct drive torque acting on the joint axis ê is given y ( ( )) B ˆ ˆ DD τ = I ω e + ω I ω e (9) CMGs actuate the root via internal momentum exchange. The total momentum of the lin and CMGs is H = h + h + h lin CMG CMG 2 ( ) ( ) H I ω I ω ω h I ω ω h G/B G2/B = + + + + + + lin gr r gr r 2 (20) The sum of the rotor momenta taes the form (Fig. 2) h + h = 2h sinφe ˆ (2) r r2 r The momentum of the gimals relative to the ody cancels ecause the gimal rates are opposite. Without loss of generality, we comine the inertias of the lin and CMGs into the inertia dyadic I. I = I + 2I (22) lin gr With this sustitution the total momentum of Eq. (20) reduces to H I ω e ˆ (23) The derivative of the angular momentum is = + 2h sinφ r i i ( ) B = + + 2h φ cosφˆ + 2h sinφˆ r r H I ω ω I ω e ω e (24) The projection of Eq. (24) onto the joint axis ê must e zero due to the asence of joint torques for a CMG actuated root. The gimal rate and the ody rate are determined y the following: ( ) B ˆ ˆ r 0 = I ω e + ω I ω e + 2h φ cosφ (25) In our simulations elow, the gimal rates are determined directly from the joint rates and accelerations. We next show an equivalent power cost for the CMGs and direct drive. Consider a single lin that rotates aout a fixed joint axis with an angular velocity θe ˆ. The direct-drive torque is τ = I θ (26) DD The power is the product of the torque and rate. P DD = I θθ (27) With the same geometry and motion for the scissored-pair actuated lin, Eq. (25) determines the gimal rate for a given lin rotation: 5
Differentiating gives an expression for gimal acceleration: I θ φ = 2h cosφ (28) r 2 I θ + 2h φ sinφ r φ = (29) 2h cosφ r For Eq. (5) we assume the gimal acceleration is small. Equation (29) shows that the ody acceleration, jer, and inertia must e small, and the gimal angle avoids saturation singularities for this to e true. The gimal torque with the angular velocity aout one axis is: τ = 2I φ 2 θh cosφ (30) A gr r The gimal-motor power is the product of the torque in Eq. (30) and the gimal rate in Eq. (28). I θ P = ( 2I φ 2 θh cosφ CMG gr r ) (3) 2h cosφ When the gimal inertia and acceleration are small, the CMG power simplifies to: P I θθ (32) CMG This is the same as the direct-drive power in Eq. (27). An interesting consequence of this result is that the effect of gimal angle in a scissored pair is removed from the power equation. A non-zero-momentum set point would not affect the power usage except as needed to maintain the torque in Eq. (30). The rotor momentum is also asent from Eq (32). One source of uncertainty in the rotor momentum is a constant gimal-angle offset due to misalignment of the gimal angles in a scissored pair. Therefore a constant gimal-angle misalignment will not affect CMG power. We also note that this result does not depend on torque amplification. Any decrease in gimal torque due to torque amplification will e offset y a decrease in velocity, and vice versa. Furthermore, leveraging torque amplification in CMGs requires a gimal rate greater than the ody rate [7]. However agile systems may have performance requirements that contradict this requirement. To show this, we use conservation of momentum aout the joint axis to determine the maximum gimal angle, φ pea. 2h sin( φ ) = I θ (33) r pea pea In other words, φ pea measures how conservative the CMGs are sized for a given system. This equation with Eq. (28) provides a ound on the gimal rate: θ pea φ tan ( φpea ) (34) θ pea r Sizing a CMG to provide torque amplification may artificially limit φ pea. However, such a system may offer other advantages in accuracy, andwidth, and motor size. This analysis demonstrates that internal momentum exchange is not inherently inefficient relative to direct drive. Using CMGs to rotate a root lin does not significantly increase the power cost of the system. The CMG system does add complexity and rotor losses to the rootic system, ut it provides the valuale opportunity for reactionless actuation. Figure 4. Lin rotation. B. Single-Lin Simulations In this section, we simulate a single lin actuated y either direct drive or a scissored pair of CMGs as discussed aove. The lin motions are prescried to 6
facilitate comparison etween direct drive and scissored pairs without confounding factors from a particular control algorithm. The lin is rotated through a given angle in least time, suject to maximum rate, acceleration, and jer requirements (see Fig. 4). We aritrarily select a maximum lin rate, acceleration, and jer of (rad/s, rad/s 2, rad/s 3 ), and the lin rotates 2 rad. The angle profile shown in Fig. 4 is achieved via the following relationships among total lin rotation and maximum rate, acceleration, and jer: 2 ω j = a (35) max max max ω a max Δ θ = ω + max max a j max max (36) Δ t = 4 a j (37) max max For a given trajectory, we calculate the torque and power required y the direct-drive and CMG actuators, and integrate power over time to otain the total energy used y each actuation method. We calculate the percent energy difference (PED) of the scissored pair relative to direct drive: E E CMG DD PED = 00 (38) E. Gimal inertia variation In the first simulation, we explore the contriution of 2I gr φ to gimal power (Eq. (3)) We vary oth the gimal inertia, I gr and the rotor momentum or equivalently φ pea. This analysis assumes that the gimal inertia does not increase the ody inertia; i.e., the sum in Eq. (22) is constant for these simulations. The only effect gimal inertia has on energy use independent of the total ody inertia is its contriution to the gimal torque of Eq. (4). The aseline parameters for all single-lin simulations are given in Tale. We used 30 evenly spaced values for φ pea and I gr over the ranges specified in Tale. The results are shown in Fig. 5 with the aseline case indicated with a white dot. It can e shown using Eqs. (28) and (29) that I gr *(cosφ) -4 enters into the expression for gimal power explaining the sharp rise in power as the CMGs approach a singularity. The flat region indicates a large design space for φ pea and I gr availale to root designers interested in CMGs. DD Figure 5. Gimal inertia and pea gimal angle effect on CMG energy use. Tale. Parameters for single-lin simulation. Δθ, deg ω max, s - a max, s -2 j max, s -3 I, g m 2 h r, m s φ pea, deg I gr, g m 2 PED, % Baseline 5 0.53 70 0..0 Gimal study " " " " " 0.50 0.59 57 86 0 0.3-0. 9 Base rate study " & 0.5 " " 0.8.2 0.44 0.63 63 78 0. ave.2 2. Transverse rate A rootic lin may e attached to a moving spacecraft ase or other rootic lins. The second simulation assumes that the spacecraft is rotating aout a transverse axis at a constant rate throughout the maneuver. The assigned ody rate is given y 7
[ θ ω ω ] T ω (39) 2 3 = where ω 2 and ω 3 are constants such that total transverse rate is less than 0.5 rad/s. We also vary the lin inertia to ensure the off-axis rotation fully contriutes to the dynamics. In a physical inertia, the maximum principal inertia cannot e greater than the sum of the other two principal inertias. The inertia matrix must also e symmetric. Pec descries a method of simulating a distriution of random inertia matrices [8] that selects the principal inertias and randomly rotates this diagonal matrix. In this wor the principal inertias are drawn from the sum of uniform distriutions to explore the parameter space, not to perform an exhaustive search. The rotor momentum is determined from conservation of momentum as per Eq. (33) using I = I ê ê and φ pea = 70 deg. The range of values of φ pea given in Tale reflects the angular momentum from the gimals and asymmetry of the inertia dyadic aout the joint axis. The results for 000 simulations are shown in Fig. 6. The total transverse rotation appears to affect the performance of the CMG root (Fig. 6a). However, y plotting the percent energy difference PED against φ pea and the ratio of gimal inertia to the ody inertia, we reproduce the asic shape of Fig. 5. We conclude that transverse rate does not affect the energy used y a scissored pair eyond its influence on φ pea and the relative sizes of the gimal and ody inertias. a Figure 6. Effect of transverse rate. a. PED vs. total transverse rate.. PED vs. φ pea and I gr /I. A root with just one joint provides several important lessons for designing CMGs for rootics. Derivations and simulation confirm that CMG actuation does not add energy costs aove those of direct-drive actuation in reasonale operating regimes. Undersized CMGs and uly gimals add to the energy costs, ut not necessarily offaxis rotations. Electrical or mechanical efficiencies including the cost of maintaining the CMG rotor at a constant speed in the presence of drag losses also add to the energy costs ut have not een considered in this study. The transverse rate of a lin appears not to directly affect energy costs, though it may influence other factors. When not operating near singularities, power is independent of the rotor s angular momentum, including gimal angle offsets, and the gimal rates. We also give a simple expression for the pea gimal rate ased on performance parameters of the root. Taen together, these results provide a straightforward means of sizing CMGs for a particular root application. IV. Multilin Root We use Kane s equations [9, 20], equivalent to the principle of virtual power in this formulation [2], to derive the equations of motion for n-lin rootic systems with either direct-drive or scissored-pair actuation. We use the following form of Kane s equations to develop the equations of motion. i v ω m v F + H M = 0 for =.. n (40) q n n a i a i i i i i i= q i= The numer of lins is n, i sums over each lin, and indexes the generalized coordinates. There are n generalized coordinates for a grounded serial linage with revolute joints. The applied forces and moments are F a and M a, with a suscript indicating the lin eing acted on. The partial derivatives in Eqs. (40) are nown as partial velocities and indicate the component of the velocity or rate aligned with the appropriate generalized coordinate. The velocities and rates are taen with respect to an inertial frame. Each ody frame is denoted Bi, or i for short, with the asis vector ê i aligned with the i th joint axis. The zero lin is taen as the nonrotating frame. We use the angle of 8
rotation of each lin aout its joint axis as the generalized coordinates, q i. We do not write one large expression for the equations of motion; rather we separate the pieces into locs that can e assemled for any numer of lins. Recursive expressions for each term allow additional arms to e added using the same loc of code with another joint-angle command. Since we use prescried motion, we do not include a feedac term. Including feedac requires an expression for the mass matrix, a simple exercise in algera and index accounting that we do not include here. A schematic of the code structure for the direct drive simulation is shown in Fig. 7. The angular velocity of lin i with respect to the ewtonian frame is defined recursively as: The angular acceleration is also given recursively: i B / = q ˆ i- i i + ω e ω (4) i i- i B / B / = q ˆ q ˆ i- + i- i i i i Figure 7. Code structure for n-lin root with direct drive. ω e e ω ω (42) From Eq. (4), the partial angular velocity term in Eqs. (40) can e concisely written. i ω eˆ i = q 0 > i (43) We let I i denote the inertia of lin i and its actuator. Differences etween direct-drive and scissored-pair inertias are identical to changing the ody inertia. We set aside the question of the relative mass of a scissored pair and a direct-drive motor and gearox ut acnowledge that it would receive some attention in a trade study of a specific application. The angular momentum of lin i and its derivative are given in Eqs. (7) and (8). Appropriate indexing of the latter equation yields: i i i i i = + i i H I ω ω I ω (44) The position vector of a single lin could easily e couched in the inertia. For an n-lin mechanism, we define l i as the vector from the i-frame origin to the i+-frame origin, and the vector from the i-frame origin to the center of mass of lin i is r i. With these definitions, the position and velocity of the i th lin relative to the inertial frame origin are written recursively as: i r = l + r (45) i/ j i j = i j i = + i/ j i v ω l ω r (46) j = The corresponding partial velocities are: v i / q = i eˆ B i j r l j= + i 0 i < (47) where it is understood that l j no longer enters the equation when i=,. The acceleration of lin i is 9
i j j j j i i i { j ( j) } i ( i) v i / = ω l + ω ω l + ω r + ω ω r (48) j = With no applied forces in the prolem, F a =0. The applied moments on lin i for direct-drive are a M = τ eˆ τ ˆ i i i i + e (49) i+ The applied moments ecome much less cumersome after summing over all the lins. The th equation from Eqs. (40) has a single torque term after taing the sum of applied moments dotted with the partial angular velocities. i a ω M = {( ˆ ˆ ) ˆ } i τ e τ i i i + e e (50) i+ i= q i= i= ω i a M = τ (5) i q The equations of motion for the direct-drive root can e assemled with a mass matrix M and the velocity product terms V. M Θ, Θ Θ+ V Θ, Θ = Τ (52) ( ) ( ) where the joint angles, rates, and accelerations and the joint torques are the elements of the arrays ΘΘΘ,,, and Τ. As with the single lin, the equations for the CMG root share most of the terms from the equations for the direct-drive root. The only differences are removing the applied joint torques and replacing them with a controlled internal-angular-momentum from the CMGs. The angular momentum of a lin and its CMGs is given y Eq. (23). The angular momentum derivative is given y Eq. (44) added to the following CMG terms: d ( 2h sinφ ) 2 cos ˆ i 2 sin ˆ r i = h φ r i φie + ω h φ i r ie (53) i dt After taing the dot product with the partial velocities, the CMG-specific terms in the equations of motion are i 2h φ cosφ eˆ eˆ + ω 2h sin φ eˆ e ˆ, for i (54) ( ) r i i i r i i The same M and V matrices can e used in the equations of motion with the addition of a vector of the gyroscopic coupling terms, B. The result is a differential equation in oth Θ and the gimal angles Φ. M ΘΘ, Θ+ V ΘΘ, + B ΘΘΦ,, = P ΘΦ, Φ (55) ( ) ( ) ( ) ( ) where the matrix P is an upper-triangular matrix that reflects the alignment of an outoard scissored pair with the inoard lin of interest. We show elow that the form of Eq. (55) as compared to Eq. (52) results in greater differences etween CMGs and direct drive than indicated y the single-lin analysis. V. Two-Lin Root The two-lin root gives insight into the expanding design considerations for multilin roots y providing a ridge etween the one degree of freedom root discussed aove and a full 3 to 6 degree of freedom root. We discuss the interactions etween the two lins and how that affects the performance of the CMGs relative to direct drive. We first introduce an analogy relating the applied CMG torque to a ody torque conceptually applied y the hand of God. This analogy illustrates a fundamental difference etween the CMGs and direct drive not seen in the single lin example. We then use simulations to find rootic motions that are particularly well suited for CMGs. A. Body Torque Analogy With direct drive actuation, each motor moves its own lin and reacts the torques produced y lins further down the chain according to the angles etween the joint axes the joint topology. For a root with parallel joint axes, 0
torque applied aout the second joint must e reacted y the first joint. The joint torques on perpendicular joint axes are independent ecause the torque is reacted against a constraint. With CMG torques, the joint axes create a zero-torque oundary condition for each lin, although torque perpendicular to the joint axes does affect neighoring CMGs. To illustrate the concept, we replace the CMGs with a Figure 8. Body torques and joint torques. phantom ody torque on each lin (Fig. 8). We use a suscript j or to denote joint torques and ody torques. The equations of motion in matrix form with joint torques are already given in Eq. (52). For the case of ody torques, we define a parameter γ i,j as the dot product of the two joint axes. γ = eˆ e ˆ (56) i, j i j The equations of motion for the ody torques can e written as: γ,2 M( ΘΘ, ) Θ+ V( ΘΘ, ) = Τ 0 (57) The matrix P in Eq. (55) is also an upper triangular matrix with the γ i,j multiplied y 2h r cosφ i. Equations (52) and (57) easily permit us to write ody torques in terms of joint torques: γ,2 Τ = Τj 0 (58) The torque relationship does not conclude the question of joint vs. ody torques. The angular velocity required to determine the power needed y each actuator also varies. Joint torques use the joint velocities the θ s used to descrie the motion of the root. The ody torques use the component of the vector ω i/ along the joint axis, determined y summing up the inoard joint velocities scaled y γ i,j. We can write the ody velocities in terms of the joint velocities for the two lin case as 0 Θ = Θ j γ (59),2 The power of the ody torques can now e easily expressed in terms of the joint torques and velocities. For a perfectly restorative system (e.g. with only springs), we can show that the power is equal in oth cases. γ,2 γ,2 P + P = Θ P P 2 j Τ = + j j j2 0 0 (60) As discussed aove, we are concerned with a nonconservative system. We sum the asolute value of the power at each actuator to otain: P + P = θ τ γ τ + γ θ + θ τ (6) 2 j j,2 j2,2 j j2 j2 Collecting terms, we write the power of the ody torques in terms of the joint torques and an extra cross term etween the rate of the first lin and the joint torque on the second lin. P + P = P γ θ τ + P + γ θ τ (62) 2 j,2 j j2 j2,2 j j2 This expression shows that joint torques and ody torques will have different power requirements and energy demands for certain maneuvers. CMGs are an excellent means of ody torque actuation, as indicated y the singlelin analysis aove, and may e used to exploit advantages availale to the ody torques. The ideas of this section also simplify the design of a multilin root that uses CMGs and explain the results in the following section. B. Two Lin Simulations
We contrast a root with perpendicular joint axes and a planar root with parallel joint axes. The first could provide orientation control with two degrees of freedom (e.g., azimuth and elevation) for pointing a camera or sensor at a target. The latter provides range for reaching tass for a manipulation root on a spacecraft. These two cases provide two extremes of Eq. (6) with γ,2 equal to 0 or (Eq. (56)).. Orthogonal joint axes When the joint axes are orthogonal, not only is γ,2 equal to 0, ut the equations of motion are decoupled when the second lin is axisymmetric aout its joint axis and the two joint axes intersect. Therefore we expect a similar power performance as for the single lin. The simulations specifically offset the joint axes, translate the centers of mass, and include offdiagonal terms in the inertia matrices to expand the results in the earlier section. The asymmetry of the second lin comines with misalignment of the axes to introduce fluctuations of the system angular momentum that affect CMG sizing. The CMGs on the inner lin must account for the angular momentum of oth the inner and outer lin taen aout the first joint axis. The CMGs on the second lin need only conserve angular momentum of the second lin aout its joint axis (ecause an inertial ase is assumed). However, the net angular velocity includes a component from the first joint velocity that can project along the second joint axis, even when the axes are perpendicular. The CMGs on a multi-lin root could e sized according to the expected maximum angular momentum aout each joint, with larger CMGs on the inner lin and smaller CMGs on the outer lin. An economic alternative is to place identical CMGs on each lin and adjust the maximum joint velocity so that the net angular momentum is ounded y the capacity of the CMGs. We opt for the latter option in this wor. We calculate the angular velocities of the joints corresponding to the maximum angular momentum aout each joint and solve for the maximum angular velocities that will not saturate the CMGs. For this exploratory study, we aritrarily set the pea acceleration and jer to the same numerical value as the pea angular velocity to satisfy Eq. (35) while limiting the contriution of maximum jer to the gimal acceleration and power cost (cf. Eqs. (29) and (4)). The angle of rotation and time to accomplish the maneuver are given y Eqs. (36) and (37). The start time of the rotations are offset y Δt 0, a parameter randomly assigned to either the firstor second-lin maneuver a Figure 9. Joint topologies. a) Perpendicular joint axes. ) Parallel joint axes. Tale 2. Parameter value ranges for two lin simulations. All other parameters use the aseline values given in Tale. Joint axes are orthogonal to each other. Energy change is the range after 000 simulations Parameters Inputs I gr, g m 2 h r, m s I,ii, g m 2 r(i), m l (i), m Δt 0, s sign(δθ) Ortho 0. 0.53 0.8.2-0.5 0.5-0 4 +/- Parallel 0. 0.53 0.8.2-0.5 0.5-0 4 +/- Ortho Parallel Intermediate calculations with the other lin starting at t=0. The parameters used for a 000 trial simulation are given in Tale 2. The principal inertias, center-of-mass offset, and joint axis locations are randomly assigned component-wise from a uniform distriution over the range given in Tale 2. We use a uniform distriution to perform an exploratory analysis, not to assign statistical distriutions to the outputs. The maximum inertia aout the joint axis, the pea ody rate, and pea gimal angle are calculated and given in Tale 2. Figure 0 plots the PED calculated for oth lins together against the pea gimal angle of the first lin, φ pea. The greater inertia aout the first joint axis causes the first lin s gimal angle to influence PED more than the second lin s gimal angle. Although not shown here, the comined inertia of oth lins affects the relative performance of CMGs. The comined inertia of oth lins varies with time and determines the angular momentum aout the first joint axis (and φ pea ). Therefore we only show PED vs. φ pea. As with the single lin root, the PED 2 Result ω max, s - I max, g m 2 φ pea, deg Δθ, deg PED, % 0. 0.5.9 6. 7 74 0 59 0.6.2 0.8.6 50 80 65 34 0 33 0.03.0.0 4.0 3 73 4-2 0.0 0.9 0.9.5 73 0.2 98-49 38
is greater than zero with some dependence on the size of the CMG as represented y the pea angle for the first gimal. The PED values for two lins are greater than for the single lin ecause of the interplay etween the increased ody inertia relative to jer (cf. Eqs. (29) and (4)). Figure 0. Relative performance of CMGs and direct drive for two orthogonal lins. Figure. The joint-angle product determines if CMGs use less energy than direct drive for parallel-joint-axes roots. 2. Parallel joint axes one of the simulations thus far have indicated a compelling advantage of CMGs over direct drive or a asic joint motor. A two lin root with parallel axes taes advantage of the fundamental differences in how torque is applied to each lin as discussed in section V.A. We continue to use identical CMGs on each lin and adjust the maximum joint rate accordingly. For parallel joint axes, the sign of each joint s rotation is critical in determining the maximum joint rate (cf. Eq. (59)). When oth joints move together, the second lin saturates its CMGs more easily ecause oth joint velocities add to determine the lin s angular momentum. When the joints move in opposite directions, the second lin can attain a high joint rate while eeping the angular momentum low. We calculate the maximum joint velocities according to the sign of the desired rotations. Zero velocity of either lin is also possile due to the start-time offset Δt 0 and may e the limiting case in selecting maximum joint velocity. For 000 trials over the same range of parameters as the orthogonal-axes case, the PED is less than zero for 403 trials. A ey indicator of the PED is the product of the joint angles (Fig. ) in particular the sign agreement of the angles. This shows that CMGs are more efficient than direct drive when the joints move in opposite directions as they must for reaching tass. Direct drive is etter when the joints move together typical for moving an item across the root s field of view. VI. Conclusion This study explores potential advantages of CMGs over direct drive as rootic actuators. We review CMG dynamics for a rootic system with a focus on the gimal torque and power required to produce a given ody rate. The analysis of the CMG dynamics also provides an expression for pea gimal rates a useful design tool for sizing gimal motors. We use a scissored pair on each rootic lin aligned with the joint axis to minimize off-axis torques and to mitigate the power costs due to off-axis ody rotation. A simple expression for the power used y the scissored pair in rotating a single ody shows that power for direct-drive actuation is equal to power for CMGs. Gimal inertia, CMG saturation singularities, and lin inertia and jer hurt performance when not carefully considered. Simulations that include transverse rotations indicate that relative power use depends on these factors rather than the transverse rotation. When sizing gimal motors, pea gimal torques and rates can e easily estimated from simple equations arising from CMG dynamics and conservation of momentum. We develop recursive equations of motion for a general n-lin root to facilitate simulation. For a two-lin root, CMGs can provide an advantage in power use ecause the exchange of momentum etween the CMGs and the ody acts as a ody torque rather than a joint torque. This advantage depends on the angle etween adjacent joint axes, with parallel joint axes preferred, and whether the joints rotate in the same or opposite directions. CMGs are etter suited for reaching radially from the ase, while direct drive is etter for panning across ones field of view relative to the ase. The scissored-pair CMG system provides reactionless actuation of a root at a power cost comparale to direct drive actuation, a promising comination for future use of CMGs in non-traditional applications. 3
Acnowledgments D. Brown was supported y an SF IGERT grant. The author also thans Dr. Mason Pec and his la at Cornell University for additional funding and equipment and many excellent discussion, with special thans to Michele Carpenter whose wor in this area oth inspired and directed my efforts. References [] Billing-Ross, J.A., and Wilson, J.F., "Pointing System Design for Low-Disturance Performance," AIAA Guidance, avigation, and Control Conference, Vol., AIAA, Washington, DC, 988, pp. CP406-444-45. [2] Schau, H., Vadali, S.R., and Junins, J.L., "Feedac control law for variale speed control moment gyros," Journal of the Astronautical Sciences, Vol. 46, o. 3, 998, pp. 307-328. [3] Wie, B., "Space Vehicle Dynamics and Control," AIAA Education Series, AIAA, Washington, DC, 2008, [4] Lappas, V.J., Steyn, W.H., and Underwood, C.I., "Attitude Control for Small Satellites Using Control Moment Gyros," Acta Astronautica, Vol. 5, o. -9, 2002, pp. 0-. [5] Wie, B., Bailey, D., and Heierg, C., "Rapid Multitarget Acquisition and Pointing Control of Agile Spacecraft," Journal of Guidance, Control, and Dynamics, Vol. 25, o., 2002, pp. 96-04. [6] Pec, M.A., "Low-Power, High-Agility Space Rootics," AIAA Guidance, avigation, and Control Conference, AIAA, Washington, DC, 2005, pp. CP6243. [7] Romano, M., and Agrawal, B.., "Attitude Dynamics/Control of Dual-Body Spacecraft with Variale-Speed Control Moment Gyros," Journal of Guidance, Control, and Dynamics, Vol. 27, o. 4, 2004, pp. 53-525. [8] Yang, L.F., Miulas, M.M., Jr, Par, K.C., "Slewing Maneuvers and Viration Control of Space Structures y Feedforward/Feedac Moment-Gyro Controls," Journal of Dynamic Systems, Measurement, and Control, Vol. 7, 995, pp. 343-35. [9] Carpenter, M.D., and Pec, M.A., "Dynamics of a High-Agility, Low-Power Imaging Payload," IEEE Transactions on Rootics, 2008 (To appear), [0] Crenshaw, J.W., "2-SPEED, A Single-Gimal Control Moment Gyro Attitude Control System," AIAA Guidance and Control Conference, AIAA, Washington, DC, 973, pp. CP895. [] Aurun, J.., and Margulies, G., "Gyrodampers for Large Space Structures," ASA, 597, 979. [2] Havill, J.R., and Ratcliff, J.W., "A Twin-Gyro Attitude Control System for Space Vehicles," Tech. Rep. ASA T D- 249, 964, [3] Brown, D., and Pec, M.A., "Scissored-Pair Control Moment Gyros: A Mechanical Constraint Saves Power," Journal of Guidance, Control, and Dynamics, to appear, [4] Lisa, D., "A Two-Degree-of-Freedom Control Moment Gyro for High-Accuracy Attitude Control," Journal of Spacecraft and Rocets, Vol. 5, o., 968, pp. 74-83. [5] Schau, H., and Junins, J.L., "Analytical Mechanics of Space Systems," American Institute of Aeronautics and Astronautics, Inc., Reston, Virginia, 2003, pp. 438-440. [6] Kuroawa, H., "Survey of Theory and Steering Laws of Single-Gimal Control Moment Gyros," Journal of Guidance, Control, and Dynamics, Vol. 30, o. 5, 2007, [7] Lappas, V.J., Steyn, W.H., and Underwood, C.I., "Torque Amplification of Control Moment Gyros," Electronics Letters, Vol. 38, o. 5, 2002, pp. 837-839. [8] Pec, M.A., "Uncertainty Models for Physically Realizale Inertia Dyadics," The Journal of the Astronautical Sciences, Vol. 54, o., 2006, [9] Kane, T.R., and Levinson, D.A., "Dynamics: Theory and Applications," McGraw-Hill, ew Yor, 985, [20] Kane, T.R., and Levinson, D.A., "The Use of Kane's Dynamical Equations in Rootics," The International Journal of Rootics Research, Vol. 2, o. 3, 983, pp. 3-2. [2] Moon, F.C., "Applied Dynamics: With Applications to Multiody and Mechatronic Systems," Wiley, ew Yor, 998, pp. 270. 4