The Kalman Filter

A scientific analogy of a spiritual truth, dedicated to Grant Rouze on his retirement from Collins Aerospace.

 


On July 2, 1937, Amelia Earhart and Fred Noonan lifted their heavily-loaded Lockheed Electra off the airfield at Lae, New Guinea. Their destination was Howland Island, 2200 miles away.  Howland was a tiny sliver of land in the vast Pacific Ocean, only a mile long and no more than 10 feet at its highest point.250px-Howland_island_nasa

Finding Howland would be no simple task, but Earhart and Noonan had a foolproof plan. Noonan was skilled in the use of a sextant, an ancient navigation tool which which multiple measurements of the sun’s height above the horizon could be ascertained to determine one’s position.  The Electra was equipped with a loop antenna attached to a Bendix radio direction finder (RDF) receiver, as well as a pair of Western Electric receiver/transmitter radios. Earhart had coordinated with the USS Itasca, which was to be stationed off Howland with instructions to transmit a homing beacon on a frequency the RDF could receive.  Itasca was also to attempt voice communications, using 3105 kHz during the night and 6210 kHz during the day.  As a final precaution, Itsaca was to ignite her oil-fired boilers to generate smoke when the Electra neared the island the following morning. 330px-Earhart-electra_10

In spite of her careful planning, Earhart’s attempts to find Howland proved fruitless.  She was never able to determine a bearing from Itasca’s transmission.  Nor did she ever receive Itasca’s radio communications. The smoke from Itasca’s boilers was never seen. Earhart’s final words are a tragic and chilling commentary to her inability to navigate with the conventional methods of her day:

“WE MUST BE ON YOU BUT CANNOT SEE YOU…BUT GAS IS RUNNING LOW…BEEN UNABLE TO REACH YOU BY RADIO…WE ARE FLYING AT A 1000 FEET”

The Self-Contained Navigation System

In navigation, knowing your precise location and that of your destination is important. Earhart was totally dependent on receiving a signal from an external signal at the desired destination.

It took several years before the dawn of the self-contained navigation system, introduced by the pioneer manufacturers at Delco who designed the first Carousel Inertial Navigation System (INS) for early versions of the Boeing 747, which was also the first inertial used for military aircraft including the KC-135, C-5A, and C-141 airlifters. INS uses motion sensors (accelerometers), rotation sensors (gyroscopes), and gravity to determine position. Then in the 1960’s and 70’s, Roger L. Easton conceived, patented, and led the development of the essential enabling technologies for the United States Global Positioning System (GPS). GPS triangulates position based on signals received from multiple satellites. The INS and GPS self-contained navigation systems could compute their position without any help from ground-based navigation aids. These were perfect solutions for navigation in previously uncharted areas.

Each of these systems has strengths and limitations. GPS repetitively computes a highly accurate, reliable position as long as the satellite signals are healthy.  However, GPS only computes its position once per second, too slow to keep up with a fast-moving airplane flying at 300 knots. On the other hand, the INS computes its position ten times per second, fast enough to keep up at high speeds. The problem with inertial is that its position estimation algorithm has a tendency to drift over time. It can drift so badly that in an hour, its position can be off by nearly a mile. If only there was a way to combine the best of these two technologies–to get a repetitively accurate position at a high rate suitable for airplane navigation.

The Kalman Filter

In the early 2000’s, a solution evolved whereby inertial and GPS position computations were combined in an estimation algorithm known as the Kalman filter, named for Rudolf E. Kalman, one of the primary developers of the theory. KalmanThe Kalman filter compares the precise position of GPS with the drifting inertial position, computing the difference and maintaining it over time in an error correction database of sorts. The Kalman filter applies the correction to the high-rate inertial output before outputting it to the pilot’s navigation system.  In this way, the Kalman filter computes a reliable, accurate, high-speed position perfect for navigation anywhere in the world.

A Spiritual Reality

It is often true that God uses physical world phenomenon to illustrate spiritual truths. He gives us pictures to illustrate what He’s trying to communicate. The Kalman filter is one of those phenomenon. It gives us a model for the way a man can be pleasing to God.  Its components demonstrate a natural law of man and the gentle, helpful nature of God. God gives us a desirable destination, one that is characterized by peace, joy, and contentment, defines a path for us to attain it, and offers the precise position updates we need to stay on course. Undoubtedly we all want to arrive at that destination, but we also seem to take issue getting there with the way He prescribed.

Man, for his part, is like the inertial. Always spinning, churning, trying to get places as fast as possible. He is remarkable in his self-sufficiency and ability, yet he is prone to wander. He often finds himself far from the intended path, hopelessly lost, unable to shake feelings of anxiety, disillusionment, and despair.

God is like the GPS. His Word provides the course corrections we need, not just once but on a regular basis. Wise is the one who is willing to see his propensity to drift and to seek and heed corrections in his daily life.

Applying the Kalman Filter to My Life

I am a big advocate of spiritual self-examination.  Like any man, I am prone to wander, always thinking the best of myself, easily blinded by my own vanity and self-deceptive heart. I need external truth sources to help me see the blind spots.

The author of Psalm 119 applies self-examination in verses 59 and 60.  He says he “considered his ways” and “turned his feet to God’s testimonies”.  He noticed he was drifting, so he put God’s truth in front of him and made a change. Perhaps he had been thoughtless or uncaring, had said an unkind word in a moment of rashness.  He acknowledged his error and immediately found the one offended and apologized.  He didn’t wait a week while he “prayed about it.” He took immediate action. This is applying a Kalman Filter of sorts to my life–it’s religion of the best kind.

Verses like Psalm 139:23-24 are vital to a daily walk with God. David, owning his inability to scrutinize himself, asks God to do it for him…

“Search me, O God, and know my heart; try me and know my anxious thoughts, and see if there be any hurtful way in me, and lead me in the everlasting way.”

I think this is what Paul is talking about in II Corinthians 13:5 when he says we are to “examine” ourselves.  We should look at the way we are living, be open to critique, and be willing to make any necessary corrections. One way to do that is to consider the message of Jesus Christ. He says He is the “Way, Truth, and Life”.  He is the standard by which all things are measured. The right response is to take Him up at His offer to “come to Me, all who are weary and heavy-laden.”  When we learn to do that over time, we begin to enjoy the benefits He promises, “…you shall find rest for your souls” (Matthew 11:18-20).

We all drift off God’s path from time to time.  It is inevitable.  All the more reason to love and appreciate the gentle, repetitive way God calls us back, like the GPS in the Kalman filter. God’s gentle correction is expressed best by the prophet Isaiah, who tells us, “…your ears will hear a word behind you, “This is the way, walk in it,” whenever you turn to the right or to the left.”

Praise God for His patience with us, and His gentle course corrections.

Post Script — Farewell to Grant Rouze

I post this blog with a bit of sadness as I say “farewell” to Grant Rouze, my long-time friend and peer at Collins Aerospace.  This month, Grant is retiring after 35+ years of service.  Grant RouzeGrant will always be known as the master of Kalman filter implementations in Collins navigation systems. The photo at left shows Grant (with fellow engineer Cris Johnson) evaluating the performance of the Collins Integrated Navigation solution during a flight test on a Canadian Air Force C-130. 

Grant and I shared some  fun memories together.  The most vivid of which was the time we almost flooded the hanger at Spar Aerospace in Edmonton, Alberta, when I nearly snapped a photo with my flash camera.  One of the mechanics caught me in time, informing us that if the light-sensing system picked up my flash, it would have fired off the giant foam guns, filling the space with 10 feet of flame-retardant foam.

Grant, may all your in-air alignments reach Nav mode!

 

 

 

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