Reference frames used in this paper are defined as follows:

The inertial frame (O_iX_iY_iZ_i). As shown in Figure 1, the inertial frame has its origin at the center of the Moon. Its z-axis is normal to the equatorial plane, x-axis is in the equatorial plane and points to the vernal equinox, and the y-axis completes a right-handed orthogonal frame.

An INS usually includes a navigation computer and an inertial measurement unit (IMU), which typically consists of three orthogonal accelerometers and three orthogonal gyroscopes. By tracking both the current angular velocity and the current linear acceleration of the explorer measured by IMU, the INS determines the linear acceleration of the explorer in the inertial frame. Thus, if the original velocity and position are known, the inertial velocity of the explorer can be obtained by integration of the inertial acceleration, and integration again yields the inertial position. There are two types of inertial navigation systems: platform inertial navigation system and strap-down inertial navigation system (SINS). In the platform inertial navigation system, IMU is mounted on a mechanical platform, which can isolate explorer’s motion and is held in alignment with the expected navigation frame. The main disadvantages of this system are that the mechanical platform is expensive and its moving parts tend to wear out or jam. In the SINS, IMU is mounted rigidly onto the explorer, and a mathematical platform takes the place of the mechanical platform. This reduces the cost and size, increases the reliability by eliminating the moving parts. In this study, SINS is used for navigation of the lunar explorer. The basic equations of inertial navigation in the navigation frame can be simply expressed as follows [6]: (1) { r ˙   =   D v v ˙   =   R b n   f b   −   ( 2 w im n   +   w mn n )   ×   v   −   g R ˙ b n   =   − Ω bn n R b n D   =   [ 1 / R m 0 0 0 1 / ( R m   cos   L ) 0 0 0 1 ]where r = [L, λ, h]^T is the explorer’s position vector, L, λ, h are explorer’s latitude, longitude and altitude in the Moon fixed frame. v = [v_x, v_y, v_z]^T is the explorer’s velocity vector. f^b is the output of accelerometers. w im n   =   [ 0 ,   w m   cos   L ,   w m   sin   L ] T is the rotation rate vector of the Moon fixed frame with respect to the inertial frame. w_m is the magnitude of the rotation rate of the Moon and has the value 2.66 × 10⁻⁶ rad/s. w mn n   =   [ − v y / R m ,   v x / R m ,   v x   tan ( L ) / R m ] T is the rotation rate vector of the navigation frame with respect to the Moon fixed frame. R_m is the radius of the Moon and has the value 1,738 km. g = [0,0,1.618m/s²]^T is the gravity vector. Ω bn n is a skew symmetric matrix which can be described by the gyro outputs w ib b and w in n The superscript n and b denote the navigation frame and the explorer body frame, respectively. R b n is the transformation matrix from the explorer body frame to the navigation frame, which is also the attitude matrix and can be defined as (2) R b n   =   [ cos   φ   cos   Ψ   −   sin   φ   sin   Ψ   sin   θ   −   sin   Ψ   cos   θ cos   Ψ   sin   φ   +   sin   Ψ   sin   θ   cos   θ cos   φ   sin   Ψ   +   sin   φ   cos   Ψ   sin   θ cos   Ψ   cos   θ sin   Ψ   sin   φ   −   cos   Ψ   sin   θ   cos   φ −   cos   θ   sin   φ sin   θ cos   θ   cos   φ ]

From the principle of INS we can see that an initialization is needed before INS can properly work. INS initialization is the process of determining initial values for position, velocity, and attitude in the navigation frame, and in some cases, inertial sensor errors are also estimated. INS attitude initialization is called alignment, which is the process of determining the initial values of the coordinate transformation from the body frame to the navigation frame in SINS [1].

As a relatively mature technology, initial alignment of INS on the Earth has been widely studied in the literature. The main research directions include: INS error models [7–9], filter methods [10,11], observability analysis [12,13], and etc. The basic principle that most methods and techniques are based on is as follows. INS is initially provided with its position from a human operator or GPS, etc. Its initial velocity can be set to zero if it starts from rest. The alignment can be accomplished by sensed gravity and Earth’s rotation vectors [14,15]. However, on the lunar surface, the accuracy of position provided by the ground tracking is only about 1,000 m. Though the gravity information is available, the lunar rotation rate is so small that its accuracy of measurement is not high enough for executing alignment. Thus, there must be other means to execute the INS initialization. It also can be seen that small measurement errors introduced by accelerometers and gyroscopes are integrated into large errors in velocity and position. These errors are cumulative and increase with time. The errors existing in inertial sensors include zero-mean random errors and systematic errors. Systematic errors, such as scale factor and bias variations can be modeled and calibrated. In this study, only biases are calibrated in the initialization.

Stars always move in the regular way, their positions can be known exactly at a specific time. Celestial navigation is a kind of technology of finding one’s position through astronomical observations. CNS is usually comprised of a star sensor (or sun sensor) and an inclinometer. The star sensor is used to measure the direction of the star and the inclinometer is used to measure local vertical direction. Thus the star altitude, which is the angle between the horizon and the line of sight to the star, can be subtended. The star altitude is a function of the explorer’s position and the geographic position (GP) of the star, which is expressed in the Moon fixed frame as follows: (3) { sin   L   ⋅   sin   Δ   +   cos   L   ⋅   cos   Δ   ⋅   cos   t LHA   =   sin   H t LHA   =   t GHA   +   λwhere H is the star altitude. Δt_LHA are the declination and local hour angle of the star. t_LHA is the sum of λ and t_GHA. t_GHA is the angular distance to the meridian of the GP measured westward from the 0° longitude. The GP of a star can be determined by its Δ and t_LHA, which can be obtained from the Astronomical Almanac together with the precise measurement time. The geometric meanings of these parameters are shown in Figure 2.

When there are enough measurements, the explorer’s position can be determined by the intercept method or the filter method [16]. It can be seen from above principle that the navigation accuracy of CNS depends mainly on the accuracy of star altitude measurements. In fact, there are many errors existing in star altitude measurements, such as sensors (for both the inclinometer and the star sensor) index error, signal noise, and alignment error. These errors can significantly degrade navigation accuracy. Previous study shows that 5.93″ is the maximum allowable lumped error that will ensure accuracy of coordinate results to within 50 m [17]. These errors can be broadly divided into two types: systematic error and random error. Systematic error is usually constant and can be modeled by mathematic functions. Random error is the unpredictable error and cannot be modeled. In this study, the systematic error in CNS measurement is estimated and corrected by the help of INS measurements using following method.

Because the lunar explorer is stationary, the state model in the Moon fixed frame is defined as: (4) X ˙   =   0 X   =   [ L ,   λ ,   θ ,   φ ,   ψ ,   b x ,   b y ,   b z ,   a x ,   a y ,   a z ,   α ] Twhere L, λ are lunar explorer’s position (latitude and longitude). θ, φ, ψ are attitude Eula angles. b_x,b_y,b_z are gyros errors; a_x,a_y,a_z are accelerometers errors; both errors are simply modeled as a constant [20]. α is the systematic error in the star altitude measurement, which also can be modeled as a constant.

To make all state variables observable, the star altitude, star orientation and outputs of IMU are chosen as the measurement variables.

Star altitude. From Equation (3), the measurement equation of star height H is obtained as: (5) H   =   arcsin ( sin ϕ   ⋅   sin   Δ   +   cos   φ   ⋅   cos   Δ   ⋅   cos   t LHA )   +   α   +   v H t LHA   =   t GHA   +   λ ϕ   =   ϕ e   +   Δ ϕ λ   =   λ e   +   Δ λwhere λ_e, ϕ_e are the pre-estimated longitude and latitude of lunar probe respectively, which is provided by ground tracking network. The error between λ_e, ϕ_e and the true value is about 1,000 m. α is the systematic error in the star altitude measurement as mentioned above and v_H is measurement noise.

The Kalman filter (KF), which is optimal for application to linear and Gaussian systems, is often used in the INS initialization [22]. However, in this celestial assisted INS initialization system, the measurement model is nonlinear. Since UKF has lower estimation errors than the Extended Kalman filter (EKF) for nonlinear systems and it also avoids the derivation of Jacobian matrices, UKF is used in this study [23,24]. The block diagram of the celestial assisted INS initialization method based on UKF is shown in Figure 3.