JP5052852B2 - Method and system for determining momentum paths without singularities - Google Patents

Description

  The present invention relates to the field of spacecraft maneuvering control, and more specifically to a method and system for determining momentum paths without singularities.

  Various rotary inertia members can be used to control the altitude of the spacecraft. One such inertia member is a control moment gyroscope (CMG). A CMG typically includes a flywheel with a constant or variable spin rate attached to a gimbal assembly. The spin axis of the CMG can be tilted by moving the CMG using a gimbal assembly. This motion generates a gyro torque that is orthogonal to the spin and gimbal axes.

  For complete spacecraft altitude control, a minimum of three CMGs can be used, with each CMG in the CMG array arranged to provide a torque centered on a primary independent axis. Typically, an additional CMG is provided to provide redundancy and to help avoid singularities. Singularities occur when the CMG momentum vectors are aligned so that they cannot provide one or more components of the required torque.

  A number of different techniques have been developed to avoid singularities. Note that in one method, Jacobian A maps the CMG gimbal velocity to a three-dimensional array torque.

Aω = τ (1)
Where A is a 3 × n Jacobian matrix, ω is an n × 1 array of gimbal velocities for n gimbals, and τ is a 3 × 1 array of torque components applied to the spacecraft. From the above equation and using the known torque command τ, the individual gimbal speed of each CMG can be calculated. Using the known Moore-Penrose pseudoinverse matrix to find the inverse of the Jacobian matrix, the set of possible gimbal velocities is ω = A ^T (AA ^T ) ⁻¹ τ (2)
It is.

As described above, the use of CMG has the potential to align the CMG momentum vector in a row such that it reaches a singularity condition. Mathematically, there is a possibility that singularities occur when the eigenvalues of AA ^T approaches ^0, approaches ^{(AA T) -1} is infinite. Or therewith but is equivalent things, singularity (expressed ^{algebraically, det (AA T) = 0} ) the determinant of the matrix AA ^T is equal to 0 occurs when. For 3 × n matrix A, which corresponds with the rank of the matrix AA ^T is 2 or less.

Different approaches have been devised to avoid singularities in CMG movement. One approach is to replace (AA ^T ) ⁻¹ with (AA ^T + εI) ^−1, where I is the identity matrix and ε is a small number, so that (AA ^T ) ⁻¹ is never zero. . By using positive ε, det (AA ^T + εI) ⁻¹ never becomes zero.

  While this approach is useful in some cases, this approach has the disadvantage of changing the gimbal velocity calculation. In the case of Jacobian A, using a pseudo inverse matrix means that the gimbal speed is no longer accurately mapped to the commanded torque because of the error ε. The resulting error can maneuver the spacecraft in the wrong direction, and significant undesirable torque can be introduced, especially near singularities.

  In the second approach, the momentum output of the CMG array is limited to a small region within the momentum envelope. This momentum envelope is the momentum given by all possible combinations of CMGs in the CMG array. Depending on the placement of the CMG, in one exemplary embodiment, singularities can be avoided by manipulating within 1/3 or less of the total momentum envelope. However, this approach wastes potential torque, resulting in a system that is larger and heavier than necessary.

  Another approach uses a steering method that avoids singularities. By the steering method, a momentum path without a singular point can be determined before moving the CMG. The problem with the steering method is that determining the path is typically an operation that involves a huge amount of computation, and as a result, a considerable amount of time is required between the issuance of a command to rotate the spacecraft and the actual start of rotation. This causes a delay. What is needed is a steering method that can quickly and efficiently determine a momentum path without a singular point.

  In one exemplary embodiment, a method for avoiding singularities in the movement of CMGs within an array of CMGs in a spacecraft is disclosed. First, it receives a maneuver command that rotates the direction of the spacecraft. The torque required to rotate the spacecraft orientation is then determined. Next, the momentum path is determined by integrating the torque. Next, the momentum path is decomposed into a line segment. For each line segment, a unit vector along the straight line segment is determined. Next, it is determined whether or not there is a continuous path connecting the start point and end point of the line segment in a plane perpendicular to the unit vector. A set of gimbal angles is determined for each point along the path in a plane perpendicular to the unit vector.

  In another exemplary embodiment, a spacecraft control system is implemented. The control system receives a steering command to rotate the spacecraft orientation, calculates the torque required to rotate the spacecraft orientation, integrates the torque to determine the momentum path, and linearizes the momentum path. It is possible to determine whether there is a continuous path connecting the start point and end point of a straight line within a plane perpendicular to the unit vector by determining a unit vector along a straight line for each line segment. Includes advanced control system. The control system further comprises a momentum actuator control processor coupled to the advanced control system. The momentum actuator control system is configured to determine a set of gimbal angles for each point along a path in a plane perpendicular to the unit vector. The control system further comprises a plurality of control moment gyroscopes coupled to the momentum actuator and configured to receive commands from the momentum actuator to rotate the spacecraft orientation.

  The present invention is described below with reference to the drawings, wherein like numerals indicate like elements.

  The following detailed description is merely exemplary in nature and is not intended to limit the invention or the application and uses of the invention. Furthermore, there is no intention to be bound by any expressed or implied theory presented in the preceding technical field, background, brief description of the drawings, or the following detailed description.

  The following detailed description describes the use of the present invention in connection with its use in an exemplary CMG sequence singularity avoidance system. However, the application of the present invention is not limited to one specific application or embodiment, and is useful in various fields of trials.

  An exemplary control system 100 that implements the present invention is shown in FIG. The components of the control system 100 are known in the art and can be assembled in various ways using different processors, software, controllers, sensors, and the like. In addition, various computing functions typically provided in one part of the system may be implemented in other parts instead. A system 100 as shown in FIG. 1 includes portions relevant to the description of the present invention, and the system 100 is well known, which can also be implemented in a control system, but FIG. Other elements or systems not shown may be included.

  The control system 100 includes an advanced control system 102 coupled to a momentum actuator control processor 104. CMG 106 is coupled to momentum actuator control processor 104. Associated with each CMG 106 is one or more CMG sensors 108 for providing information about the state of the CMG 106 to the control system. In one embodiment, the control system 100 is mounted on a space probe such as an orbiting satellite.

  The altitude control system 102 controls the positioning of the spacecraft. The altitude control system 102 receives data relating to the desired spacecraft maneuver and determines an appropriate torque command to complete the desired maneuver. Torque commands can be presented to the momentum actuator control processor 104. The momentum actuator control processor 104 can calculate the gimbal speed required to generate the commanded torque in response to the torque command. Further, the momentum actuator control processor 104 can calculate the gimbal movement from the momentum path determined by the steering method.

  Based on the above calculation, the momentum actuator control processor 104 gives the necessary command to the CMG 106 so that the torque commanded by the CMG movement is generated, and the torque of the momentum actuator control processor 104 avoids the singularity according to the teaching of the present invention. give.

  FIG. 2 avoids singularities, and in this embodiment, in a momentum space where corresponding paths in a set of four CMGs aligned on the plane of a four-sided pyramid are prepared in a gimbal angled space. 6 is a flow diagram of an exemplary embodiment of a method 200 for preparing a straight line path. Three CMGs are required to have three degrees of freedom, and the fourth CMG can be used for redundancy and to help avoid singularities.

In step 202, which is the first step, the three-dimensional momentum space formed by the movements of the four CMGs is decomposed into a two-dimensional projection. First, the momentum given for all four CMGs is
h = h ₁ + h ₂ + h ₃ + h ₄ (3)
Please note that.

However, each subscript represents an individual CMG. In this method, the commanded momentum path is a straight line h (t) = h ₀ + t * w (4)
It's above.

Here, h ₀ is a point on the momentum path , t is a scalar quantity (for example, a time parameter) that moves along a straight line, and w is a unit vector along the direction of the straight line.

Assuming Equations 3 and 4 are equal,
h ₁ + h ₂ + h ₃ + h ₄ = h ₀ + t * w (5)
It becomes.

  Since w is a unit vector along a straight line, w_perp can be defined as a projection onto a plane orthogonal to w. For example, if w is a unit vector that moves in the x axis, w_perp is the y and z planes.

Multiplying both sides of Equation 5 by w_perp,
w_perp ′ * (h ₁ + h ₂ ) = w_perp ′ * (h ₀ − (h ₃ + h ₄ )) (6)
Is obtained because w_perp ′ * w = [0, 0].

Equation 6 represents a two-dimensional projection of the CMG momentum envelope. In the projection, both h ₁ + h ₂ and h ₀ − (h ₃ + h ₄ ) are projected onto the w_perp plane. Since Equation 4 requires that all momentums follow a straight line, when the momentum path moves along a straight line in the w plane, the movement of h ₁ + h ₂ in the w_perp plane is the w_perp plane. Must be offset exactly by the movement of h ₃ + h ₄ . Therefore, the region where the projection of h ₁ + h ₂ overlaps with the projection of h _0- (h ₃ + h ₄ ) in the w_perp plane is a possible region where the momentum path exists along the straight line w, and in the two-dimensional space The pure momentum of the constant w_perp ′ * h ₀ is perpendicular to w.

As an example of the projection of h ₁ + h ₂ onto the w_perp plane, the momentum h ₁ of the _first CMG is on a circle and the momentum h ₂ of the _second CMG is on another circle. Please keep in mind. In this exemplary embodiment, the relative speed of the CMG flywheel is such that h ₁ + h ₂ together can form a donut shape commonly referred to as a torus. Referring to FIG. 3, the torus 302 represents a momentum space formed by h ₁ + h ₂ . In FIG. 3, the torus 302 is projected onto a two-dimensional momentum envelope 304 (XZ plane in FIG. 3) to form a momentum projection 303. A vertical straight line can penetrate the torus 302 at 0, 2, or 4 locations. For example, if the first straight line 306 is outside the torus 302 or passes through a hole in the torus 302, the first straight line 306 intersects the torus 302 at zero points. The second straight line 307 intersects the surface of the torus 302 at two points 308 and 310. In addition, the third straight line 309 enters the torus 302, passes through two points, then continues through the hole and exits on the opposite side of the torus 302, after which the straight line is further 2 in all four locations. Penetrate the torus at one point. For example, the third straight line 309 can intersect the torus 302 at point 312, point 314, point 316, and point 318.

W_perp * (h ₁ (θ ₁ ) + h ₂ (θ ₂ ) in order to determine the number of points where the straight line can intersect the momentum envelope given an arbitrary point (y, z) in the projection plane ) = [Y, z] can be used to solve for the angles θ ₁ and θ ₂ . Solving these equations gives a 4th order polynomial with four solutions on the complex number field. Neglecting the complex (non-physical) solution, we conclude that there can be 0, 2, or 4 real solutions corresponding to the number of points where the straight line intersects the momentum envelope in 3D space.

Referring to FIG. 4, FIG. 4 illustrates a projection 400 of w_perp * (h ₁ + h ₂ ) for a particular CMG arrangement. Projection 400 includes an outer boundary 402 and an inner boundary 404. These boundaries separate different solution regions of w_perp * (h ₁ (θ ₁ ) + h ₂ (θ ₂ )) = [y, z]. There are four actual solutions in the first region surrounded by the inner boundary 404. There are two real solutions in the second region 408 between the inner boundary 404 and the outer boundary 402. There is no actual solution outside the outer boundary 402.

FIG. 5 shows an exemplary projection 500 of w_perp ′ * (h ₀ − (h ₃ + h ₄ )). These boundaries separate different solution regions of w_perp * (h ₀ − (h ₃ (θ ₃ ) + h ₄ (θ ₄ ))) = [y, z]. In FIG. 5, there is an outer boundary 502 and an inner boundary 504 that intersects itself at two endpoints. There are two actual solutions in the first region 506 between the outer boundary 502 and the inner boundary 504. There is no real solution in the second region 508, which is relatively elliptical and surrounded by the inner boundary. There are four real solutions in the third region 510, surrounded by the inner boundary 504, which is somewhat triangular.

Of course, as noted above, the region of interest is the overlap of the two projections. FIG. 6 shows the projection of FIG. 4 and the projection of FIG. The first region 602 represents a region where solutions for the w_perp ′ * (h ₁ + h ₂ ) projection and the w_perp ′ * (h ₀ − (h ₃ + h ₄ )) projection overlap. Since each h _i draws an edge centered on the origin, w_perp ′ * (h _i + h _j ) is centered on the origin, and w_perp ′ * h ₀ is the center and center of the two donut-shaped regions. Is the distance between. In particular, the first region 602 represents the overlap of the second region 408 of the projection 400 and the first region 506 of the projection 500. In regions where there is no overlap, such as the second region 604, or where 0 solution regions overlap 2 or 4 solution regions, such as the third region 606, there is no solution. The fourth region 608 represents the overlap of the first region 406 of the projection 400 and the first region 506 of the second projection 500. The fifth area 610 represents the overlap of the second area 408 of the projection 400 and the third area 510 of the second projection 500.

Since different regions have different numbers of solutions, when moving along a momentum path , if the path starts in a region with more solutions than the number of solutions in the region where the path stops in the w_perp plane, the wrong path will be When initially selected, there can be no continuous solution between the two regions, and it may be necessary to select a different path to reach the end point. This is an example of a gimbal lock inside the momentum space.

Returning to FIG. 2, in step 204, when the boundary curves for the w_perp ′ * (h ₁ + h ₂ ) and w_perp ′ * (h ₀ − (h ₃ + h ₄ )) regions are constructed, the first of the roadmap A part is generated. A boundary curve is defined as a boundary that separates regions with different numbers of solutions. Mathematically, these can be found by letting h _{i be} the angular momentum of the i th CMG and ‖h _iと す る be a constant momentum magnitude. Each momentum vector h _i moves along a circle. If e _i is a constant unit vector perpendicular to the plane containing the circle, e _i '* h _i = 0.

If the momentum path h = h ₀ + w * t is given, the momentum perpendicular to the straight line must be 0, so w_perp ′ * (h ₁ + h ₂ + h ₃ + h ₄ ) = w_perp ′ * h ₀ But because

That's why. This means that w_perp ′ * (h ₁ + h ₂ ) = w_perp ′ * (h ₀ − (h ₃ + h ₄ )). Since each h _i moves along a circle, a combination of two of these gives a two-dimensional region. A two-dimensional region swept out by [y, z] = w_perp ′ * (h ₁ + h ₂ ) in a plane perpendicular to w is bounded by two curves. These two one-dimensional curves are cases where the function of the two variables is singular, so instead of giving a two-dimensional region, the function degenerates to a one-dimensional boundary curve. A bivariate function is singular when the 2 × 2 Jacobian matrix is singular. The function [y, z] = w_perp ′ * (h _i + h _j ) has a Jacobian matrix [dy, dz] = w_perp ′ * (dh _i + dh _j ), where dh _i = e _i × h _i is a vector perpendicular to the h _i in the direction of h _i changes when the h _i are moved along the circle. The Jacobian is singular if and only if there is a two-dimensional unit vector v _ij that the Jacobian sends to 0 (and thus v _ij is on a circle), ie v _ij '* w_perp' * [e _i × h _i e _j × h _j ] = [0 0] (7).

Similarly, the function [y, z] = w_perp ′ * (h ₀ − (h ₃ +... + H _n )) is the Jacobian matrix [dy, dz] = w_perp ′ * (h ₃ +... + Dh _n ), Which is the two-dimensional unit vector v that Jacobian sends to 0 _{. . n} is _unique (and thus v _3..n is on a circle) and is only singular, ie v _{3. . . n} '* w_perp' * [e ₂ × h ₃ . . . e _n × h _n ] = [0. . . 0] (8).

However, three vectors a, b, the c, using a vector identical equation a '* a (b × c) = det [ a b c] a = w_perp * v ij, and b = _{e i,} and c = _{h i} Each side of the above equation can be converted to:

Therefore, h _i is orthogonal to e _i × (w_perp * v _ij ), h _i is orthogonal to e _i , and thus h _i has the direction of the outer product of two vectors. Thus, the unit vectors in these directions are equal until there is a change in sign, as follows:

  Multiplying this expression by w_perp gives the following expression:

  However, in this last equation, the following vector identities including arbitrary vectors b and c and unit vector e were used.

The 2 × 2 symmetric matrix A _i can be defined as follows.

A _i = w_perp ′ * (I−e _i * e _i ′) * w_perp (13)
The following expression is obtained with use of this matrix by the formula of the last w_perp '* _{h i.}

The inner and outer (below using + and-signs) boundary curves of the w_perp ′ * (h ₁ + h ₂ ) region are obtained by moving v ₁₂ on a circle and calculating the following equation:

The boundary curve inside and outside of the w_perp ′ * (h ₀ − (h ₃ + h ₄ )) region (below using the + and − signs) is calculated by moving v ₃₄ on the circle and calculating can get.

The 2 ⁿ⁻³ boundary curves (below all code selections are used) of the w_perp ′ * (h ₀ − (h ₃ +... + h _n )) region are v _{3. . . It} is obtained by moving _n on the circle and calculating the following equation.

When calculating curves for the w_perp ′ * (h ₁ + h ₂ ) and w_perp ′ * (h ₀ − (h ₃ + h ₄ )) regions, the intersection points of the curves are for later use in the roadmap calculation. Saved and the cusp of the curve is also saved.

Referring to FIG. 7, FIG. 7 illustrates another arrangement of four CMGs with w_perp projections of h ₁ + h ₂ and h ₀ − (h ₃ + h ₄ ). In this _embodiment, the value of _{h 1, h 2, h 3} , and _{h 4} are different as Figures 4-6. The projection of w_perp ′ * (h ₁ + h ₂ ) is represented by a solid line, and the projection of w_perp ′ * (h ₀ − (h ₃ + h ₄ )) is represented by a broken line. There are four areas. The first region 702 has a total of 16 solutions, the second region 704 has 8 solutions, the third region 706 has 4 solutions, and the fourth region 708 has Has 0 solutions. As can be seen, the first boundary curve 710 separates the first and second regions, the second boundary curve 712 separates the second region and the third region, and the third boundary curve 714 is A third region 706 and a fourth region 708 are separated, and a fourth boundary curve 716 separates the fourth region 708 from points outside the projection.

  The boundary curve functions as a path or path in the projection. Movement from one region to the other can be accomplished by moving along the boundary curve until the other boundary curve or intersecting road or path is reached. In FIG. 6, the boundary curves intersect. In FIG. 7, the boundary curves do not intersect. This does not mean that there is no path from one region to the other region in FIG. Within the roadmap it is implied that additional roads are needed.

  At step 206, additional roads for the roadmap are determined by constructing horizontal tangents that connect the boundary curves. The tangent is configured to be a tangent to the boundary curve. Since the tangent is horizontal, it becomes the tangent of the boundary curve at the maximum and minimum values. In FIG. 7, an exemplary horizontal tangent line 720 that intersects the boundary curve is illustrated.

  Further, a horizontal line passing through the cusp of the boundary curve is drawn. A cusp is defined as a point where the direction of the boundary curve changes suddenly. As shown in FIG. 7, the first boundary curve 710 has a first cusp 722, a second cusp 724, a third cusp 726, and a fourth cusp 728. A cusp line 730 passing through the cusp is drawn.

The cusp at which the curve suddenly changes direction can be calculated by noting that the formula for the boundary curve of two dimensions [y, z] = w_perp ′ * (h ₁ + h ₂ ) is given by the following equation: it can.

In order for a curve to suddenly change direction, all components of its derivative must be zero. To compute the derivative of the right side of the equation with respect to two-dimensional unit vector parameter v _12, it is necessary to evaluate the following equation.

  Since v '* v = 1, the change in v must be perpendicular to v as follows:

Therefore,

  The formula (v ′ * A * v) * A−A * v * v ′ * A of the 2 × 2 matrix in the above formula is

and

Can be simplified by multiplying all scalar terms by

Is obtained.
Using this identity in the previous equation and paying attention to ‖v‖ ² = 1 and (l−v * v ′) * (I−v * v ′) = (I−v * v ′) The following equation is obtained.

For both A ₁ and A _2, using this type of expression, from the part of the curve equation, the conditions of curve cusps of the derivative of the curve occurs when the 0 is obtained.

Therefore, the scalar multiplied by the matrix (I−v ₁₂ * v ₁₂ ′) must be zero. The outer boundary curve corresponds to the + sign in the above equation, and the sum of the two positive terms cannot be zero, and therefore the outer boundary curve cannot have a cusp. The inner boundary curve corresponds to the minus sign in the above equation, and a cusp occurs in the inner boundary curve if the following holds:

Multiplying the above equation by (v ₁₂ '* A ₁ * v ₁₂ ) ^3/2 * (v ₁₂ ' * A ₂ * v ₁₂ ) ^3/2 gives the following equation:

  When both sides are raised to the 2/3 power, the following formula is obtained.

c _i = Position and (‖h _i ‖ ^{_{* det (A i)) 2/3}} , then the following equation is obtained when out factoring the right side _{v 12} and the left side of the _{v 12} '.

_{0 = v 12 '* (c} 1 * A 2 -c 2 * A 1) * v 12 (29)
v To solve this equation for _12, the eigenvalue decomposition of 2 × 2 symmetric matrix

And

Then, the cusp equation is as follows.

Therefore, this equation has a solution if and only if λ ₁ and λ ₂ have opposite signs. Assuming that λ ₁ is the maximum eigenvalue, if λ ₂ is negative, the four cusp solutions for the unit vector v ₁₂ are as follows:

Similarly, the cusp on the inner boundary curve of the w_perp ′ * (h ₀ − (h ₃ + h ₄ )) region is
0 = v ₃₄ '* (c ₃ * A ₄ -c ₄ * A ₃ ) * v ₃₄ (33)
Is given by the solution of

  After the cusps are determined, horizontal lines through those cusps can be drawn to form additional paths connecting the boundary curves.

Similarly, when expanding to N CMGs, the cusp on the inner boundary curve of the w_perp ′ * (h ₀ − (h ₃ +... + H _n )) region is

Is given by the solution of

(V _{3... N} ′ * A ₃ * v _{3... N} ) ^3/2 *. . . * (V3 _{... n} '* _An * v3 _{... n} ) When multiplied by ^3/2 , the following equation is obtained.

If both sides of such an expression are raised to the 2/3 power, rearranged, raised to the 2/3 power again, and this process is repeated n-3 times, the unit vector v3 _{. . . A} very high degree polynomial (on the order of 2 ⁿ⁻² ) is obtained in the ratio of the two components of ⁿ . Forming and solving such a high degree polynomial is very difficult for n> 4, the cusp of the boundary curve,

Instead, v3 _{. . .} When performing numerical calculation of a curve when _n draws a unit circle, it should be calculated with care when the direction of the curve changes suddenly.

With these additional “roads”, to move from one boundary curve to the other, move along the boundary curve until it reaches the tangent, and then move along the tangent to the other boundary curve To do. In addition to the straight line constructed in the last step, additional straight lines are needed to connect the start and end points with boundary curves, tangents, and cusp lines. Thus, at step 208, the horizontal line 743 is constructed across all boundary curves from the start point 740 and the horizontal line 745 is constructed across all boundary curves from the end point 742. Since the momentum path is a straight line in a plane orthogonal to the w_perp plane, the projection of start and end points coincide. However, although straight lines exist on different layers of projection, the start and stop points are at different distances along the w line, so they are somewhat similar to double-layer bridges or highways. For example, returning to FIG. 3, if the start point coincides with the point 312 and the end point is at the point 318, the two points are clearly different points in the three-dimensional space, but once projected into the two-dimensional space. And the start point and end point seem to match.

  After all the paths have been constructed, the intersection of the straight line and the curve can be calculated at step 210. An intersection represents a case where the solution path can change from one region of the solution to the other region of the solution. Therefore, the intersection behaves like an expressway exit and an entrance ramp to the road. For example, the starting point 740 starts from the first region 702. In order to move to the second region 704, a movement can be made from the straight line 743 to the first boundary curve 710 with an intersection. A movement can then be made along the first boundary curve 710 to where one of the horizontal tangents 720 intersects the boundary curve. At this intersection, movement into the second region 704 can be made along the horizontal tangent line 720.

  Once roads and intersections are constructed in two dimensions, step 212 needs to determine which of those intersections are intersections in three-dimensional space. The intersection in the three dimensions, the associated boundary curve, and the horizontal line then represent the roadmap.

  Next, in step 214, an algorithm such as Dijkstra's algorithm can determine the best path to take from start to finish. Dijkstra's algorithm developed an algorithm for finding the shortest path from the first point on the graph (start point 740) to the second point on the graph (end point 742). W. Named after Dijkstra. In this algorithm, intersections are considered vertices in the graph and boundary curves and horizontal curves are graph edges.

  FIG. 8 illustrates a method 800 for controlling a spacecraft in accordance with the teachings of the present invention. First, in step 802, a steering command for rotating the direction of the space probe is received. In one exemplary embodiment, the maneuver command is transmitted from the ground control station to the spacecraft altitude control system 102. Alternatively, the maneuver command can be generated by the spacecraft based on a pre-planned travel schedule.

  After the steering command is received, at step 804, torque as a function of time required to maneuver the spacecraft is calculated in the control system 100. This can be performed in the advanced control system 102 in one embodiment.

  In step 806, the torque calculated in step 804 is integrated to determine a momentum path as a function of time. This calculation can be performed in the advanced control system 102. At step 808, the momentum path is broken down into a series of straight lines. Next, for each individual line segment, it can be determined whether there is a continuous path from the start point to the respective end point of the line segment. The number of line segments required to approximate the momentum path depends on how accurately the original smooth momentum curve needs to be approximated by the series of line segments.

  At step 810, a unit vector w is determined for each line segment. As described above, the momentum of a line segment can be expressed as follows using a unit vector.

h (t) = h ₀ + t * w (37)
t is a parameter such as time, w is a unit vector, and h ₀ is the head point of the line segment.

Next, in step 814, a roadmap algorithm is used to determine whether the path connecting the start and end points of the line segment is in the w_perp plane. Then, at step 816, if there is a path for each point along the path in the w_perp plane, the individual momentum vectors h ₁ and h ₂ for each line segment can be determined.

Given values for a point [y, z] in the plane perpendicular to w, the equation [y, z] = w_perp ′ * (h ₁ + h ₂ ) is used to determine the individual momentum vectors h ₁ and h _{1 2} can be solved. The set of equations that need to be solved are:

[Y, z] = w_perp ' * (h 1 + h 2) h 1 and two linear equations _{h 2 ‖h 1} ‖ ^{_{2 = h 1' * h 1}} h 1 quadratic ‖H ₂ ‖ ² = h ₂ '* h ₂ h ₂ quadratic equation (38)
e ₁ '* h ₁ = 0 h ₁ linear equation e ₂ ' * h ₂ = 0 h ₂ linear equation y, z, w_perp, ‖h ₁ ‖ ² , ‖h ₂ ‖ ² , e ₁ , and e ₂ Is known, so the above six equations can be solved for the six unknown components of the vectors h ₁ and h ₂ . Since four of the six equations are linear equations, they can be used to erase the four variables and leave two quadratic equations for the two remaining variables. Using standard techniques such as synthesis, two quadratic equations with two variables can be used to obtain a single quaternary equation (order 4) with one variable. Solving for the four roots of the quartic equation, then back substitution can be used to solve for the remaining five variables. This gives four solutions for vectors h ₁ and h ₂ .

Similarly, if a value is given for a point [y, z] in a plane perpendicular to w and an arbitrary value is given for momentum h ₅ to h _n , the equation [y, z] = w_perp ′ * (h ₀₋ (h ₃ + h ₄ + h ₅ +... + H _n )) can be used to solve for the individual momentum vectors h ₃ and h ₄ . The set of equations that need to be solved are:

[Y, z] = w_perp ′ * (h ₀ − (h ₃ + h ₄ + h ₅ +... + H _n )) Two linear equations of unknown h ₃ and h ₄ ‖h ₃ ‖ ² = h ₃ ′ * h ₃ h ₃ quadratic equation ‖h ₄ ‖ ² = h ₄ '* h ₄ h ₄ quadratic equation (39)
e ₃ '* h ₃ = 0 h ₃ linear equation e ₄ ' * h ₄ = 0 h ₄ linear equation y, z, w_perp, ‖h ₃ ‖ ² , ‖h ₄ ‖ ² , e ₃ , e ₄ , Since h ₅ to h _n are known, the above six equations can be solved for the six unknown components of vectors h ₃ and h ₄ . Since four of the six equations are linear equations, they can be used to erase the four variables and leave two quadratic equations for the two remaining variables. Using standard techniques such as synthesis, two quadratic equations with two variables can be used to obtain a single quadratic equation with one variable. Solving for the four roots of the quartic equation, then using backward substitution, the remaining five variables can be solved. This gives four solutions for vectors h ₃ and h ₄ .

The above method for finding a solution avoids the use of trigonometry. Another approach for finding a solution is to write h ₁ and h ₂ as the sum of a constant vector multiplied by the sine and cosine of the gimbal angles θ ₁ and θ ₂ , then the two quadratic equations (cos (θ ₁ )) ² + (sin (θ ₁ )) ² = 1 and (cos (θ ₂ )) ² + (sin (θ ₂ )) ² = 1 together with the above four linear equations.

Finally, the momentum vector can be sent to the momentum actuator control processor 104 to determine the appropriate gimbal movement. If the values for θ ₁ and θ ₂ are determined at step 816, the gimbal angle can be changed at step 818 based on the values determined for each point in the path in the w_perp plane.

  While at least one embodiment has been presented in the foregoing detailed description, it will be appreciated that a vast number of variations exist. It will also be appreciated that the example or embodiments are only examples, and are not intended to limit the scope, applicability, or configuration of the invention in any way. Rather, the foregoing detailed description will be a convenient roadmap for implementing one or more embodiments for those skilled in the art. It will be understood that various changes may be made in the function and arrangement of elements without departing from the scope of the invention as defined in the claims and their legal equivalents.

2 is a block diagram illustrating an exemplary CMG control system in accordance with the teachings of the present invention. FIG. 4 is a flowchart of a roadmap algorithm used in CMG path planning in accordance with the teachings of the present invention. FIG. 6 illustrates a torus-shaped momentum envelope projected onto a plane, according to an illustrative embodiment of the invention. FIG. 6 illustrates a projection of momentum envelopes for w_perp ′ * (h ₁ + h ₂ ) in accordance with the teachings of the present invention. FIG. 6 illustrates a projection of momentum envelopes for w_perp ′ * (h ₀ − (h ₃ + h ₄ )) in accordance with the teachings of the present invention. FIG. 6 illustrates FIGS. 4 and 5 combined in accordance with the teachings of the present invention. FIG. 6 illustrates the formation of roads and intersections within the projection of the momentum envelope of a CMG array in accordance with the teachings of the present invention. 6 is a flow diagram illustrating a method of maneuvering a spacecraft in accordance with the teachings of the present invention.

Explanation of symbols

100 control system 102 advanced control system 104 momentum actuator control processor 108 CMG sensor 302 torus 303 momentum projection 304 two-dimensional momentum envelope 306 straight line 308, 310, 312, 314, 316, 318 point 309 straight line 400 projection 402 outer boundary 404 inner boundary 406, 408 region 500 projection 502 outer boundary 504 inner boundary 506, 508, 510 region 602, 604, 606, 608, 610 region 702, 704, 706, 708, 710, 712, 714, 716 region 710 boundary curve 720 horizontal tangent 722, 724, 726, 728 Point 730 Point line 740 Start point 742 End point 743 Horizontal line 745 Horizontal line

Claims (3)

A method for avoiding singularities in the movement of multiple CMGs (106) in a spacecraft,
Receiving a command to adjust the orientation of the spacecraft;
Calculating a torque required to satisfy the condition of the command;
Integrating the torque to determine a momentum path;
Approximating the momentum path with a plurality of straight line segments;
For each segment of the plurality of straight line segments,
Determining a unit vector along the straight line;
Determining whether there is a continuous path connecting the start and end points of the line segment in a plane perpendicular to the unit vector by calculating a plane w_perp plane perpendicular to the unit vector ;
Determining a set of momentum vectors for each line segment when there is a continuous path ; and
Calculating a gimbal movement necessary for the CMG (106) for the set of momentum vectors determined for each of the segments,
Including methods. Determining whether there is a continuous path connecting the start and end points of the line segment in a vertical plane;
Projecting a three-dimensional momentum space onto a two-dimensional plane to form a w_perp projection;
Determining a boundary curve of the w_perp projection that separates regions having different numbers of solutions ;
Connecting the boundary curve with a series of horizontal tangents, each of the horizontal tangents being one tangent of the boundary curve, extending through the boundary curve; the horizontal tangent being perpendicular to the unit vector;
Wherein the boundary curve knot and cusp linear series of horizontal, each of the cusp straight line becomes tangent to the one cusp of the boundary curve, and a step to extend through the boundary curves,
Configuring a start line intersecting the w_perp projection from the start point to connect the start point and end point to the boundary curve, the horizontal tangent line and the cusp line ;
Configuring an end line intersecting the w_perp projection from the end point to connect the start point and end point to the boundary curve, the horizontal tangent line and the cusp line ;
Selecting a location where the boundary curve, the horizontal tangent line , the cusp straight line, the start line, and the end line intersect as an intersection;
And determining an optimal path from the start point to the end point using the intersection. A space probe control system (100) including an altitude control system (102) comprising:
The advanced control system (102)
Receiving a command to change the orientation of the spacecraft;
Calculate the torque required to rotate the direction of the spacecraft,
Integrating the torque to determine the momentum path;
Approximating the momentum path with a plurality of straight line segments,
For each segment of the plurality of straight line segments, determine a unit vector along the straight line segments,
Determining whether there is a continuous path connecting the start and end points of the line segment in a plane perpendicular to the unit vector by calculating a plane w_perp plane perpendicular to the unit vector;
An advanced control system (102) operable to determine a set of momentum vectors for each of the plurality of line segments when there is a continuous path;
A momentum actuator control processor (104) coupled to the altitude control system (102) and operable to determine a set of gimbal movements for each momentum vector calculated for each of the line segments;
A plurality of control moment gyroscopes (106) coupled to the momentum actuator (102) and operable to receive commands from the momentum actuator (104) to change the orientation of the spacecraft;
A control system comprising:

Images