Exercise: Simple and Combined Plane Changes (Using Targeter)

Note: To do this exercise you will need a valid license of STK Professional Edition and STK/Astrogator.

The object of this exercise is to transfer a satellite from a low-Earth, circular parking orbit with a radius of 6570 km and an inclination of 28 deg to a geosynchronous orbit with a radius of 42,160 km and an inclination of zero. One way to accomplish this is to use a Hohmann Transfer to increase the radius of the orbit, followed by a maneuver to decrease its inclination. As you'll see, it's more economical to combine the second phase of the Hohmann transfer with the inclination change into a single, combined maneuver. We'll try it both ways and consider some other alternatives.

Reference: This exercise is based on an example discussed in Sellers, Jerry Jon, Understanding Space: An introduction to Astronautics, New York: McGraw-Hill (1994), pp. 191-192.

See the technical notes for equations that demonstrate the greater efficiency of a combined maneuver at apogee of the transfer orbit.

Setup

  1. Create a scenario and a satellite.
  2. On the Orbit page of the satellite's Basic properties, select the Astrogator propagator. You may need to expand the properties window to see all of the controls.

Hohmann Transfer Followed by Plane Change

First let's try a Hohmann Transfer to achieve a radius of 42,160 km, then, after the Hohmann Transfer is completed, a separate maneuver to carry out the plane change, i.e., the change in inclination from 28 deg to zero. The MCS should appear roughly as follows when you finish constructing it:

Define the Initial State

  1. The default MCS that appears when you display the satellite's Orbit page probably already begins with an Initial State segment. If not, insert one at the beginning of the MCS.
  2. Name the segment '28 deg Inclined Orbit'.
  3. Select Keplerian as the Element Type and set the Semimajor Axis to 6570 km.
  4. Set Inclination to 28 deg. All other elements should be set to zero.

Propagate the Parking Orbit

  1. If the second segment of the MCS is not already a Propagate segment, insert one in that position.
  2. Name the segment 'Propagate 2 Hours' and, if you wish, select a different color for the segment.
  3. Set the Duration (Trip value) to 2 hours (7200 sec), more than enough to have the satellite orbit one complete pass.

Start the Hohmann Transfer

Now use the targeter to calculate the ΔV required to increase the radius of the orbit to 42,160 km. No change in inclination will be attempted in this maneuver.

Define a Target Sequence
  1. Insert a Target Sequence segment.
  2. Name the Target Sequence segment 'Begin Hohmann'.
  3. Nest a Maneuver in the Target Sequence.
  4. Name the nested Maneuver segment 'DV1'.
Select Variables
  1. Highlight the nested Maneuver and make certain that the Maneuver Type is set to Impulsive.
  2. Select Thrust Vector for Attitude Control.
  3. Select Cartesian as the vector type.
  4. Select VNC(Earth) Thrust Axes.
  5. Select the X component as the sole independent variable.
  6. Click Results... and select Radius of Apoapsis (Keplerian Elements folder) as the only dependent variable.
Set up the Targeter
  1. Select the Target Sequence, highlight the default Profile (Differential Corrector), and open its Variables page by clicking Properties....
  2. Select the Use options under Control Parameters and Equality Constraints.
  3. Set the Desired Value for Radius of Apoapsis to 42160 km.
  4. Display the Convergence page, increase the Maximum Iterations amount to 50, and select the Display Status option. Click OK to close the Properties window for the Profile.
  5. Set the Mode for the Profile to Iterate.
  6. Make sure the targeter is turned on (select Run active profiles in the Action field).

Propagate to Apogee

  1. Insert another Propagate segment after the Target Sequence.
  2. Name the segment 'To Apogee' and select a color that will distinguish it from the first Propagate segment.
  3. Insert an Apoapsis Stopping Condition and remove Duration.

Finish the Hohmann Transfer

Here you will use the targeter to calculate the ΔV required to circularize the orbit, i.e., change its eccentricity to zero. Again, no change in inclination will be targeted.

Define a Target Sequence
  1. Insert another Target Sequence segment.
  2. Name the Target Sequence segment 'Finish Hohmann'.
  3. Nest a Maneuver in the Target Sequence.
  4. Name the nested Maneuver segment 'DV2'.
Select Variables
  1. Highlight the nested Maneuver and make certain that the Maneuver Type is set to Impulsive.
  2. Select Thrust Vector for Attitude Control.
  3. Select Cartesian as the vector type.
  4. Select VNC(Earth) Thrust Axes.
  5. Select the X component as the sole independent variable.
  6. Click Results... and select Eccentricity (Keplerian Elements folder) as the only dependent variable.
Set up the Targeter
  1. Select the Target Sequence, highlight the default Profile, and open its Variables page by clicking Properties....
  2. Select the Use options under Control Parameters and Equality Constraints.
  3. Leave the Desired Value for Eccentricity at its default value of zero.
  4. Display the Convergence page, increase the Maximum Iterations amount to 50, and select the Display Status option. Click OK to close the Properties window for the Profile.
  5. Set the Mode for the Profile to Iterate.
  6. Make sure the targeter is turned on (select Run active profiles in the Action field).

Propagate to Ascending Node

To carry out a plane change to zero inclination, the satellite must be at ascending or descending node. Let's propagate it to ascending node.

  1. Insert a Propagate segment after the previous Target Sequence.
  2. Name the segment 'To Ascending Node' and select a color that will distinguish it from the other two Propagate segments.
  3. Insert Ascending Node as the sole Stopping Condition.
  4. Set the Repeat Count to 2 so that the satellite will make at least one complete orbit pass (and one will be drawn in the 3D Graphics window) before the plane change.
  5. Remove the Duration Stopping Condition.

Perform an Inclination Change

Finally, you will use the targeter to maneuver the satellite into an orbit with an inclination of zero.

Define a Target Sequence
  1. Insert another Target Sequence segment.
  2. Name the Target Sequence segment 'Simple Plane Change'.
  3. Nest a Maneuver in the Target Sequence.
  4. Name the nested Maneuver segment 'DV3'.
Select Variables
  1. Highlight the nested Maneuver and make certain that the Maneuver Type is set to Impulsive.
  2. Select Thrust Vector for Attitude Control.
  3. Select Cartesian as the vector type.
  4. Select VNC(Earth) Thrust Axes.
  5. Select the X and Y components as independent variables.
  6. Click Results... and select Inclination (Keplerian Elements folder) as the only dependent variable.
Set up the Targeter
  1. Select the Target Sequence, highlight the default Profile (Differential Corrector), and open its Variables page by clicking Properties....
  2. Select the Use options for both independent variables under Control Parameters and the dependent variable under Equality Constraints.
  3. Leave the Desired Value for Inclination at its default value of zero.
  4. Display the Convergence page, increase the Maximum Iterations amount to 50, and select the Display Status option. Click OK to close the Properties window for the Profile.
  5. Set the Mode for the Profile to Iterate.
  6. Make sure the targeter is turned on (select Run active profiles in the Action field).

Propagate the Outer Orbit

  1. Insert a Propagate segment after the previous Target Sequence.
  2. Name the segment 'Propagate 36 Hours' and select a color that will distinguish it from the other three Propagate segments.
  3. Select Duration as the Stopping Condition and enter a trip value of 36 hours (129600 sec).

Run the MCS and Compute Total Delta V

Run the MCS and observe the targeting process as displayed in the Status window. Select the 3D Graphics window, and observe that the orbit and plane transfers are distinct procedures.

Note: If the Propagate segments do not display in the selected colors, open the MCS Options window and make certain that the Draw Trajectory in 3D as it is Calculated and Use Trajectory Segment Colors options are selected.

The orbit is circular before and after the plane change, but only the final orbit appears so, since it is equatorial (has an inclination of zero) and the Earth is seen in a polar perspective.

Suggestion: Save the scenario. You'll need this MCS for the exercise below.

To compute the total ΔV for the Hohmann Transfer and plane change, select each Target Sequence segment, click Properties... and note the Final Value assigned to each Control Parameter.

Warning: Do not click Apply Changes, since that will prevent you from clearing the current targeting results for the next part of the exercise.

The values you find should be approximately as follows:

Segment Variable Value
Begin Hohmann X 2.4540 km/sec
Finish Hohmann X 1.400 km/sec
Simple Plane Change X -0.3350 km/sec
Y -1.4451 km/sec

Note: The current values you observe may differ slightly from those shown here, depending, e.g., on the Tolerances you use for the dependent variables in each Target Sequence.

The ΔV required for the plane change is given by:

Thus, the total ΔV is:

ΔVT = 2.4540 + 1.4000 + 1.4834 = 5.3774 km/sec

As shown in the technical notes, in terms of the ΔV required, this combination of maneuvers, in which a simple plane change is carried out at apogee of the transfer orbit, is less expensive than one in which the plane change occurs at perigee, but more expensive than one in which the plane change is combined with the second burn of the Hohmann Transfer. The latter alternative is considered below.

Combined Plane Change

Now let's try combining the plane change with the second burn of the Hohmann Transfer. One way to do this is to constrain the second Target Sequence in terms of both inclination and eccentricity -- i.e. equatorialize and circularize the orbit in the same maneuver.

Redesign the MCS

Instead of building a new MCS from scratch, use cut-and-paste to adapt the one you just created:

  1. Highlight each target sequence, in turn, and click Reset Profiles.
  2. Delete the first Propagate segment ('Propagate 2 Hours').
  3. Move the next-to-last Propagate segment ('To Ascending Node') to a position immediately after the Initial State segment ('28 deg Inclined Orbit').
  4. Select the first of the nested Maneuver segments (DV1) and rename it 'Simple DV'.
  5. Delete the middle Target Sequence ('Finish Hohmann'). Its nested Maneuver will automatically be deleted at the same time.
  6. Select the final Target Sequence ('Simple Plane Change') and rename it 'Combined Change'.
  7. Select the final nested Maneuver segment ('DV3') and rename it 'Combined DV'.

When you are finished, the MCS should look like this:

Adjust the Targeting Parameters

Now make sure the variables are set correctly:

  1. Select the first of the nested Maneuver segments ('Simple DV') and make certain that the X (Velocity) component is selected as the independent variable and that Radius of Apoapsis is selected as the dependent variable.
  2. In the first Target Sequence ('Begin Hohmann'), open the Variables page for the default Profile (Differential Corrector) and make certain that the Use option is selected for both variables, and that the Desired Value for the dependent variable, Radius of Apoapsis, is set to 42160 km.
  3. Select the second of the nested Maneuver segments ('Combined DV'), select the X and Y components as independent variables, and select Eccentricity and Inclination as dependent variables.
  4. In the second Target Sequence ('Combined Change'), open the Variables page for the default Profile (Differential Corrector), and make certain that the Use Option is selected for both independent variables and for both dependent variables, and that the Desired Values for Eccentricity and Inclination are set to zero and 0 deg, respectively.
  5. Increase the Maximum Iterations value (on the Convergence page) for the second Target Sequence to 100.

Run the MCS and Compute Total Delta V

Clear the ephemeris of the previous run, and then run the MCS again. Select the 3D Graphics window and observe that the orbit and plane transfers occur as part of the same procedure.

Again, look at each targeting profile and observe the Final Values assigned to the control variables. The ΔV for the combined eccentricity and inclination change is:

The total ΔV for the transfer from the low-Earth, inclined parking orbit to the equatorial outer orbit is:

ΔVT = 1.8286 + 2.4570 = 4.2856 km/sec

which is less than the amount calculated above for a Hohmann Transfer followed by a separate plane change. Why should this be so? See the technical notes. Again, the current values you observe may differ slightly from those shown here, for the reasons noted above.

Maximizing the Efficiency of Plane Changes

As demonstrated above, if a plane change is to be carried out at the apogee of a Hohmann Transfer orbit, it is more efficient, in terms of the ΔV required, to use a single maneuver combining the plane change with the second burn of the Hohmann Transfer. This can also be demonstrated more formally.

It is also more efficient, in general, to carry out the plane change at apogee rather than at perigee, whether or not it is combined with the Hohmann Transfer. Thus, the most efficient approach (of those considered here) is a combined maneuver at apogee.