TERRAIN-RELATIVE PLANETARY ORBIT DETERMINATION Kevin Peterson1, Heather Jones1, Corinne Vassallo2, Allen Welkie3, and William “Red” Whittaker1 1The Robotics Institute, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213, USA 2The Department of Physics, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213, USA 3The Department of Engineering, Swarthmore College, 500 College Ave, Swarthmore, PA 19081, USA ABSTRACT enable fully autonomous operation from launch to land- ing. This paper presents an approach to visual navigation for Deep Space 1 demonstrated visual navigation far from a spacecraft during planetary orbit. Spacecraft orbital pa- planetary surfaces by triangulation to distant asteroids. rameters are determined autonomously by registration to These observations relied on the asteroid appearance as a terrain. To accomplish this, individual camera images are single pixel in an imager, reducing measurements to dis- registered to pre-existing maps to determine spacecraft criminating a point source from black background. Vi- latitude and longitude. Using knowledge of orbital dy- sual navigation during terminal descent has been shown namics, the time history of sensed latitudes and longi- in terrestrial demonstrations to be more accurate than ra- tudes is fit to a physically realizable trajectory. Outlier dio navigation and operates with latency independent of measurements are rejected using robust estimation tech- distance to Earth. With new high-resolution Mars and niques. Simulation results using data from the Lunar Re- Moon imagery, terrain relative navigation can achieve connaissance Orbiter show that method determines the 30-meter localization precision. However, autonomous orbit semi-major axis with an average error less than 6km visual-only navigation is undeveloped for planetary ap- for a 500km altitude lunar orbit. proach, capture, and orbit. Key words: orbit determination; vision; robotics. This paper describes Terrain Relative Planetary Orbit De- termination (TROD), an approach to visual navigation for orbiting spacecraft. TROD matches terrain appearance to on-line imagery and utilizes knowledge of orbital dynam- 1. INTRODUCTION ics to determine position, velocity, and orbital parame- ters of the spacecraft trajectory. Simulation results using data from the Lunar Reconnaissance Orbiter show that Visual navigation for precise, reliable guidance of crewed the method determines the semi-major axis with an aver- and robotic planetary missions offers the opportunity to age error less than 6 km for a 500 km lunar orbit. revolutionize space exploration by improving targeting of landing sites, substantially reducing mission risk, and extending human reach through autonomous missions to 1.1. Related Work distant locations such as the Jovian moons. State-of- practice radio-based navigation exhibits unacceptable la- tency at long range, relies on interaction with Earth, and Orbit determination has a deep history dating back cen- cannot achieve the landing accuracy possible through vi- turies to early physicists. The related work is deep and sual means. Autonomous navigation via camera offers it is out of scope to discuss most aspects of orbit deter- an alternative to radio without the mass and power re- mination here. Rather, we focus on spacecraft orbit de- quired for traditional altimetry (e.g., RADAR and laser), termination and computer vision approaches to trajectory and without communication to Earth. Visual navigation modeling. will enable autonomous spacecraft to land in radio-dark locations like craters and the far sides of planets. In deep space, camera-based navigation measures an- gles to distant asteroids to triangulate location [RBS+97, Visual navigation provides three primary advantages over BRK+02]. These observations rely on the asteroid ap- alternatives: accuracy, low latency, and autonomy. Dur- pearance as a single pixel in an imager, reducing mea- ing descent, accuracy and latency are important factors in surements to discriminating a point source from black mission success. Far from earth, visual navigation is crit- background. Objects that appear larger than a few pixels ical, as communication latency precludes precision ma- introduce significant angular error. The Voyager missions neuvering. Visual navigation for all mission phases will to Jupiter, Saturn and beyond used optical measurements along with radio to Earth for navigation. Images of the planet’s natural satellites against a star background were used, with knowledge of ephemerides to update a naviga- tion filter run on Earth [CSB83]. Significant work has been done on camera-based terrain relative navigation. While this technique has proved to be quite successful in simulation and sounding rocket test- ing [TMR+06, TMR+07, PBC+10], the focus has mostly been on the final stages of descent to a planetary surface, not on higher altitude trajectories. Singh and Lim imple- ment an Extended Kalman Filter (EKF) to track space- craft position and velocity in orbit; they use vision-based measurements, but they also assume that altitude can be obtained from an altimeter with only 12.5m (1σ) error. Such high-end radar is programmatically costly. They use crater-based feature tracking and thus they limit their “experiments to segments of the trajectory that span the Figure 1: TROD matches images obtained in orbit to higher latitudes where the craters are more clearly visi- maps and then applies Kepler’s laws to determine orbital ble” [SL08]. parameters. Given time and estimated latitude and longi- tude at several locations, the orbit is fully determined. In addition to planetary applications, work has also been done on autonmous navigation for landing on asteroids. The Near Earth Asteroid Rendezvous (NEAR) mission 2.2. Orbit Determination to the asteroid Eros was the first to use optical tracking of craters for navigation, but these craters were manually Kepler’s laws of orbital motion state that the area swept chosen [MC03]. The Hayabusa spacecraft, which con- by the line between a body and its orbit center in a given ducted a sample return mission to the asteroid Itokawa, amount of time is equal throughout an orbit [Sid01]. Ke- used manual tracking of landmarks for localization dur- plerian orbits are elliptical in shape with a focal point at ing descent [KHK+09]. Hayabusa also used a long range the barycenter of the system. Consequently, orbits are (50 m to 50 km) LIDAR and a short range laser range uniquely determined by several measurements of latitude, finder to determine its altitude [KHK+06]. Li presents longitude, over time as depicted in Fig. 1. a feature-based navigation routine for asteroid landing [Li08]. This method provides position and orientation Elliptical orbit of a spacecraft about a planetary body is state, although this is also defined relative to the land- described by the six classical orbital elements: marks, not with respect to a global map. A proposed method for autonomous orbit determination • a: semi-major axis for the NEAR is described in [MC03]. The reliance of • e: eccentricity this approach on the detection on circular crater features • i: inclination is somewhat limiting. As noted in [SL08], craters are • !: argument of periapsis hard to detect when the sun angle is high, since there are no shadows to define the edges of the crater rim. In ad- • Ω: right ascension of the ascending node (RAAN) dition, craters appear different at different altitudes, mak- • M: mean anomaly ing them unsuitable for use over the range of altitudes at which a spacecraft localization is needed in planetary The parameters a and e describe the size and shape of the landing missions. ellipse. The parameters i, ! and Ω describe the orienta- tion of the ellipse in space, (See Fig. 2). M describes where the spacecraft is in the orbit, although the related parameter θ, or true anomaly, is often used. 2. METHOD Orbit determination is performed in two stages. First, inclination and RAAN are determined by fitting a plane to unit vectors that extend from the orbit center towards 2.1. Overview the spacecraft. Once the plane has been determined, the noisy unit vectors are projected onto the plane to deter- mine spacecraft angular motion over time. Kepler’s laws TROD works by matching imagery obtained in orbit with describing the relationship between angular motion over orthorectified maps of the planetary surface to determine time are inverted using non-linear least squares to deter- latitude and longitude over time. Kepler’s laws are then mine a and e. The argument of periapsis is then deter- applied to determine orbital parameters. mined from the other five parameters and the measure- Z n^ is chosen to agree with the average normalized cross product of adjacent unit vectors. RANSAC is used to re- i ject spurious measurements. θ The measurement coordinate system is defined by setting the z^m axis equal to n^ and aligning the y^m axis with the cross product of n^ and r^0. The x^m axis completes the ω right-handed coordinate system. Y Ω z^m =n ^ (6) y^m =n ^ × r^0 (7) x^m = n^ × r^0 × z^ (8) X The inclination of the orbit, i, is defined as the angle be- tween the orbital and equatorial planes: Figure 2: Classical orbital elements: i, the inclination, T z^ = [0; 0; 1] (9) is the angle between the normal of the orbital plane and e T the z-axis; Ω, the RAAN, is the angle between the x-axis i = arccos(^zmz^e) (10) and the line of nodes, or the intersection between the xy- plane and the orbital plane; !, the argument of periapsis, Ω is calculated by taking the cross product of z^e and z^m is the angle between the line of nodes and the periapsis; to get the ascending node line vector, l. The ascending θ, the true anomaly, is the angle between a vector to the node line vector is then projected onto both the x^ and periapsis and a vector to the spacecraft’s current position. m y^m axes to determine Ω.