Secure motion control of micro-spacecraft using semi-homomorphic encryption

This paper studies the secure motion control problem for micro-spacecraft systems. A novel semi-homomorphic encrypted control framework, consisting of a logarithmic quantizer, two uniform quantizers, and an encrypted control law based on the Paillier cryptosystem is developed. More speciﬁcally, a logarithmic quantizer is adopted as a digitizer to convert the continuous relative motion information to digital signals. Two uniform quantizers with diﬀerent quantization sensitivities are designed to encode the control gain matrix and digitized motion information to integer values. Then, we develop an encrypted state-feedback control law based on the Paillier cryptosystem, which allows the controller to compute the control input using only encrypted data. Using the Lyapunov stability theory and the homomorphic property of the Paillier cryptosystem, we prove that all signals in the closed-loop system are uniformly ultimately bounded. Diﬀerent from the traditional motion control laws of spacecraft, the proposed encrypted control framework ensures the security of the exchanged data over the communication network of the spacecraft, even when communication channels are eavesdropped by malicious adversaries. Finally, we verify the eﬀectiveness of the proposed encrypted control framework using numerical simulations.


Introduction
Relative motion control of spacecraft is an enabling technology for many current and near-future space missions, such as orbital rendezvous, on-orbit assembly, formation flying, and reconnaissance and surveillance, which has received widespread attention in recent years [1][2][3]. These space missions usually involve an active spacecraft (chaser) and a passive spacecraft (target), and require the chaser to perform orbital maneuvers to track the specified position of the target or a virtual desired position. As the recent trend of space missions moves from human intervention towards autonomous operation, it is necessary to improve the autonomy, safety, and security of spacecraft control systems. In recent years, the autonomous relative motion control of spacecraft has received considerable attention from both academia and aerospace industrial sectors. Numerous control methods have been proposed in the literature, such as adaptive control [2], sliding mode control [3,4], model predictive control [5], and distributed control [6].
With the prosperity of fast-integrated technology, light-weight spacecraft exhibit enormous popularity, such as CubSats [7] and plug-and-play satellite [8]. Compared with the traditional monolithic and complex encrypted control framework is proposed to protect the security and privacy of intra-system signals over the communication network whether from the sensor to controller or controller to actuator. A logarithmic quantizer and a uniform quantizer are introduced, the former of which is used to quantize the continuous system states (i.e., the relative position and velocity information), while the latter is utilized to pre-process the control gain matrix and the digitized system states before encryption. Due to the homomorphic properties, the encrypted controller is constructed only using the encrypted system signals rather than the actual measurable information, which ensures a secure exchange of sensitive information among different units of the spacecraft. This paper is organized as follows. The preliminary knowledge about the dynamical model of spacecraft relative motion, Paillier cryptosystem, and logarithmic and uniform quantizers is provided in Section 2 along with the control objective of this paper. The main results of the encrypted control frame are presented in Section 3. Numerical simulations are carried out to demonstrate the effectiveness of the proposed encrypted control scheme in Section 4. Finally, some concluding remarks are given in Section 5.

System model and problem formulation
In this section, some standard notations are first defined. Then, the relative motion dynamics of the chase spacecraft in the Local-Vertical-Local-Horizontal (LVLH) frame is characterized as Clohessy-Wiltshire-Hill (CWH) equation and further transformed into the form of the general linear system. Later, Paillier encryption, logarithmic and uniform quantizers are successively introduced in order to ensure the security and privacy of the system signals over the communication network. Finally, the encrypted control problem for spacecraft proximity operations is formulated.

Notations
Throughout this paper, let R, Z, and Z * represent the sets of real numbers, integers, and non-negative integers, respectively. Z * n = {z ∈ Z : 0 ≤ z < n} defines the set of non-negative integers less than n. R n and R n×m are the sets of n-dimension vectors and n×m-size matrices, separately. The operators gcd(a, b) and lcm(a, b) stand for the greatest common divisor and the least common multiple of a ∈ Z * \{0} and b ∈ Z * \{0}. Besides, given a vector v or a matrix M , the correlated Euclidean norm for v or the induced 2-norm for M is indicated by x or M . Furthermore, given a symmetric matrix M = M , its maximum and minimum eigenvalues are denoted by λ max (M ) and λ min (M ), respectively. In addition, "mod" means modulo operation.

Dynamical model of spacecraft relative motion
Without any loss of generality, two typical reference coordinate systems, i.e, Earth-Centered-Inertial (ECI) and LVLH coordinate frames, are introduced first to describe the relative translation motion of the chase spacecraft with respect to the target spacecraft, as depicted in Figure 1. The ECI coordinate frame is denoted as O = {O − XY Z}, where its origin locates in the Earth center, X-axis points toward the vernal equinox, Z-axis is parallel to the rotational direction of the Earth and points to the north pole, and Y -axis lies in the equatorial plane and completes the orthogonal dextral frame. Let P = {o − xyz} represent the LVLH coordinate frame, which is fixed at the target spacecraft. Moreover, the x-axis in the LVLH frame is the direction of the radius vector of the target from the Earth center, the z-axis coincides with the orbital normal direction, and the y-axis completes the orthogonal dextral frame. As shown in Figure 1, denote ρ = [ρ x , ρ y , ρ z ] ∈ R 3 and r c ∈ R 3 as the position vector of the chase spacecraft in the LVLH frame and the position vector of the target spacecraft in the ECI frame, respectively. Here, it is assumed that the relative distance between the chase spacecraft and the target spacecraft is far smaller than the relative distance of the target spacecraft with respect to the Earth, that is, ρ r c . Meanwhile, the target spacecraft is supposed to move in a circular orbit. Then, the linearized CWH equation can be used to describe the relative motion of the chase spacecraft in the LVLH frame [4], which is given byρ where m c is the mass of the chase spacecraft, u = [u x , u y , u z ] ∈ R 3 denotes the control force acting on the chase spacecraft, ω = µ c /r 3 c is the mean orbital angular velocity, µ c stands for the geocentric gravitational constant of the Earth, and r c = r c .
Further, the dynamical equation (1) can be simplified in the following form oḟ The linear dynamical equation written by (2) will be exploited in the subsequent analysis. In particular, given the circular orbit of the target spacecraft, it is easy to certify that (A, B) is controllable. Hence, the linear continuous system (2) is stabilized by using the following state-feedback control law where K ∈ R 3×6 denotes the control gain matrix. For ensuring the stability of (2), K should be properly selected so that the eigenvalues of (A − BK) are in the left half-plane of the complex plane. A simple method for computing K is to pose the controller design as the linear quadratic regulator (LQR) problem [1].

Lemma 1.
[26] Consider the linear system (2) with the state-feedback controller (4). If the matrix pair (A, B) is controllable, for given any positive-definite matrix Q = Q > 0 ∈ R 6×6 , there always exists a symmetric positive-define matrix P ∈ R 6×6 such that

Paillier cyrptosystem
The Paillier cryptosystem, a typical partially HE encryption scheme, is a probabilistic asymmetric algorithm for public key cryptography [27]. The detailed realization of the Paillier encryption scheme is summarized as follows [28]: • Key generation -Select two large and independent prime numbers p and q randomly and ensure gcd(pq, (p − 1)(q − 1))=1; -Calculate the public key (N, g), where N = pq and g ∈ Z * N 2 is a random integer; -Calculate the private key (λ, µ), where λ = lcm(p − 1, q − 1) and µ = λ −1 mod N ; • Encryption -Let m ∈ Z * N be a plaintext message; -Choose r randomly such that 0 < r < N and gcd(r, N ) = 1; -Compute the ciphertext message of m as Benefiting from the additively homomorphic property and non-deterministic encryption, Paillier cryptosystem possesses the following novel features [28]: Property 1: The sum of the plaintext messages m 1 and m 2 can be calculated by decrypting the product of their corresponding ciphertext messages Enc(m 1 ) and Enc(m 2 ), which is formulated mathematically as Property 2: The product of the plaintext messages m 1 and m 2 can be determined by decrypting the product of a ciphertext message Enc(m 1 ) or Enc(m 2 ) raising to the power of a plaintext message m 2 or m 1 , which is formulated mathematically as Property 3: Consider a more general case, the product of a plaintext message m and a constant k will be computed by decrypting the product of the ciphertext message Enc(m) rasing to the power of k, which is formulated mathematically as

Quantizer
In this subsection, the logarithmic quantizer q l (·) and uniform quantizer q u (·) are discussed. To be specific, the logarithmic quantizer in this paper is utilized as a digitizer to scale the state information x in (2) so that x can be transformed into a digital signal capable of transmitting over the communication network. Meanwhile, since the Paillier cryptosystem is only able to encrypt the positive integers, the digital signals in the system should be mapped to the appropriate positive integers. Considering the quantization property of the uniform quantizer, it is not difficult to see that it can be decomposed into an encoder part and a decoder part to pre-process the digital signals before encryption.
(1) Logarithmic quantizer Referring to [29], the logarithmic set of quantization levels is defined by where ρ ∈ (0, 1) is a positive constant representing the quantization density of the logarithmic quantizer. Then, the static and time-invariant logarithmic quantizer is described as where σ = (1−ρ)/(1+ρ). It is noted from (9) that every element of the quantization level is closely related to the segment ( In this case, the logarithmic quantizer is able to map the entire segments to the quantizer level. Since the logarithmic quantizer satisfies the sector-bound condition, the quantizer q l (x) is also written as where ∆ l ∈ [−σ, σ]. Based on (10), the quantization error associated with q l (x) is defined as where ∆ l = diag{∆ l1 , . . . , ∆ ln } and I is an identity matrix with appropriate dimensions.
(2) Uniform quantizer Given a positive integer q m , the uniform quantizer is defined by [30] q where ∆ x > 0 denotes the sensitivity of the uniform quantizer and q m represents the saturation value of the uniform quantizer. From (11), it is clearly analyzed that if x ∈ ((k − 1)/2∆ x , (k + 1/2)∆ x ] where k ∈ Z and −q m ≤ k ≤ q m , then q u (x) will takes on the value k. The quantization error associated with which satisfies |x| ≤ ∆ x /2. Similarly, for any vector x ∈ R n or any matrix M ∈ R n×m , it follows that Remark 1. Since q u (x) ∈ Z, (11) can be applied as an encoder, where the real number x is encoded to an integer. Otherwise, ∆ x q u (x) is regarded as the decoder, where the encoded q u (x) is restored to approximate its original real number x.

Control objective
This paper focuses on the encrypted control problem for spacecraft proximity operations. Since the control gain matrix K can be determined offline, it does not convey sensitive information. Differently, the realtime relative position and velocity information are extremely sensitive due to some unexpected leakage or eavesdropping involved in the communication network. Hence, the relative state information x should be concealed from the controller side before encryption, which implies that the state-feedback control law in (4) can not be calculated directly by using x. In light of this, the purpose of this manuscript is to develop a Paillier-type encrypted control framework for (1), including a digitizer for continuous sampled state x and an encoder and a decoder for quantized state and control gain, and a Paillier-type encrypted state-feedback controller to achieve the following objectives: (1) ensure the ultimately uniformly bounded stability of the whole closed-loop system; (2) preserve the security of the state x from the controller.

Main results
In this section, an encrypted control algorithm by means of the Paillier encryption scheme is presented for spacecraft proximity operations. The detailed design procedure is summarized as follows: First, by using the uniform quantizer, the control gain is encoded to an integer and sent to the controller side for ease of further encryption operation. Meanwhile, the continuously measurable system states are quantized as digital signals by using the logarithmic quantizer because the communication network only allows the digital signals to be transmitted instead of the continuous ones. Similarly, for further encryption, the digitized system stats are encoded to the integer set with the aid of a uniform quantizer. Then, to guarantee data security, the system states are encrypted based on the Paillier cryptosystem and are sent to the controller side over the communication network. Next, by utilizing received encrypted system states and control gain, the encrypted control law is calculated according to the homomorphic properties in (6)- (8). Further, the resulting encrypted control law is sent to the system side without any information leakage. After Paillier-type decryption and simple decoding, a decrypted state-feedback control command is executed on the actuator of the spacecraft. The block diagram of the proposed encrypted control system is shown in Figure 2, where the solid line and dash line indicates the information exchange through the data buses or wireless communication network, respectively.

Encoding the control gain matrix
Since the Paillier encryption scheme only works the data in the form of integers, the control gain matrix K in (4) should be first encoded into an integer before being sent over the communication network, as depicted in Figure 2. As stated in Remark 1, the uniform quantizer can be regarded as an encoder and a decoder. Therefore, the uniform quantizer is adopted here to encode the control gain matrix K into q u (K). The corresponding quantizer error for K here is defined asK = K − ∆ K q u (K), where ∆ K > 0 refers to the sensitivity associated with the quantizer q u (K). For ease of convenience, definē K = ∆ K q u (K).
Theorem 1. Consider the general linear system (4) with the control law u = −Kx. Under Lemma 1, if the sensitivity ∆ K is selected such that the inequality holds trues, then the general linear system (2) is asymptotic stable, where P and Q are positive-define matrices satisfying (5).
Proof. By implementing the quantized state-feedback control law u = −Kx, the closed-loop linear system (2) is rewritten asẋ Chose V = x P x as the Lyapunov function candidate, where P is a positive-definite matrix satisfying (5). Then, combining (5) and (16) yieldṡ To proceed, it can be seen from (14) that K ≤ ∆ K 3 √ 2/2 always hold trues. Therefore, once ∆ K is chosen such that (15) satisfies, the following inequality always hold −λ min (Q) + 2 P BK ≤ −(1 − ε 1 )λ min (Q) ≤ 0 (18) which directly implies thatV ≤ 0 (19) In summary, as long as ∆ K is selected by (15), it is straightforward to induce from (19) that the linear system (2) is asymptotic stable despite the quantization. This completes the proof of Theorem 1.
Additionally, on the basis of Theorem 1 and the standard Lyapunov stability theory, there exist symmetric positive-define matricesP andQ such that

Encrypted control law design
Let x k , q u (x k ), E(q u (x k )) represent the system state after digitization, encoding, and encryption, respectively. Then, using semi-homomorphic encryption, we design the encrypted control law as follows Theorem 2. Consider the general linear system described in (2) with the encrypted control law (21). Select the quantization parameters properly such that σ < λ min (Q) 2 (22) holds, where σ is a positive constant related to the logarithmic quantizer q l (x). Then, under the proposed encrypted control scheme, summarized in Algorithm 1, the general linear system (2) is ultimately uniformly stable. Moreover, the security of sensitive information transmitted over the spacecraft's communication network is completely protected. Figure 2, it is observed that x k is sampled by the logarithmic quantizer q l (x) in (9). So, one can get that the digitization error ∆ l x ≤ σ x , where ∆ l ∈ R 6×6 is the sensitivities of the logarithmic quantizer. Besides, q u (x k ) is obtained by using the uniform quantizer q u (x) with the sensitivity ∆ x > 0. Accordingly, letx k = x k − ∆ x q u (x k ) represent the quantizer error. After decryption and decoding for the encrypted control law (21), the control command u acting on the actuator is given by

Remark 2. From
Proof. It is noted from Remark 2 that the implementation of the encrypted control law (21) is equivalent to executing u in (23). In light of this, driven by the encrypted control law (21), the closed-loop linear system (2) is rewritten asẋ Algorithm 1. The proposed encrypted control scheme for spacecraft proximity operations.
Initialize: choose p, q, g, r, K, P , Q,P ,Q Ensure: u 1: # Encode control gain matrix 2: select ∆K so that (15)  Similarly to Theorem 1, chose V = x P x as the Lyapunov function candidate, whereP is a symmetrical positive-definite matrix satisfying (20). Then, taking the differentiate V over time along (23) where x k = q l (x), ∆ l x = x k − x, andx k = x k − ∆ l q u (x k ) are used in (24). Besides, reminding (11) and (14), it is easy to obtain that Then, inserting (20) and (26) into (25), it follows thaṫ Further, recalling the parameter condition in (23), it haṡ To proceed, it is clearly seen from (28) thatV < 0 when x evolves outside of the following set It is concluded from (29) that once the system state moves in the out of S x , it will be attracted back to S x immediately. Therefore, the closed-loop system is ultimately uniformly bounded stable. Moreover, the convergence set S x can be made enough small by choosing ∆ x as small as possible and choosing σ as big as possible. This completes the proof of Theorem 2.
Remark 3. Following Figure 2, the information sent from the sensor to the controller has been encoded and encrypted before transmission over the communication network. Moreover, it is obviously seen from (21) that the encrypted control law is computed by using the encrypted relevant relative state E(q u (x k )) rather than the real-time measured value of sensor x. Therefore, there is also no sensitive relative state information that is leaked when the encrypted controller is transmitted from the controller to the actuator over the communication channel. Besides, exploiting the properties of the Pailliar cryptosystem, the proposed encrypted framework can secure the communication network of the spacecraft against false data injection attacks. For example, if the attacker injects a random real-valued noise, the controller will easily detect the false data injection attack as the transmitted signals under the Paillier scheme are integer-valued. When the attacker injects an integer-valued random noise into the communication channel, the attack cannot be detected by the controller. However, a slight change in the ciphertext results in a significant change in the computed control input (after decryption) due to the highly nonlinear operations of the Paillier cryptosystem. In this case, the actuator can detect the false data injection attack by comparing the current control input with the past control input. Consequently, the security of the spacecraft's relative motion control system is successfully protected.
Remark 4. Note that the encrypted control framework is designed based on the linear CWH equation and additive homomorphic encryption of Paillier. Although the linear CWH equation can clearly describe the relative motion between two spacecraft, a more precise control scheme can be generated by resorting to the more detailed nonlinear dynamical model. Besides, although the security and privacy of the sensitive relative motion information and control input signal are preserved, the proposed encrypted control scheme does not guarantee the security of the control gain matrix due to the inherent limitation of the Paillier encryption scheme. Therefore, the secure motion control scheme of micro-spacecraft based on the nonlinear dynamical model deserves to be investigated in future work, especially using the fully homomorphic encryption method.

Simulations
In this section, numerical simulations are performed to verify the performance of the encrypted control framework proposed in Figure 2.   implementation of the Paillier cryptosystem is referred to [31]. More specifically, two large prime numbers p and q are selected as p = 3470023813 and q = 3315231457; the random integer g in the public key is selected as g = 15314181315756238939627282471832570258; the private key (λ, µ) are selected as λ = 958661007884285856 and µ = 1687557451384170425. The simulation results are shown in Figures 3-6. More specifically, the evolution of the relative position and relative velocity of the chase spacecraft is illustrated in Figure 3. Figure 4 depicts the quantization error of the relative position and velocity for the chase spacecraft after digitization using the logarithmic quantizer. Figure 5 displays the error of the relative position and velocity of the chase spacecraft after encoding and decoding using a uniform quantizer. From Figures 3 to 5, it is concluded that although the quantization errors exist, the chase spacecraft under the proposed encrypted control framework is still able to achieve the desired tracking mission with acceptable accuracy. The time history of the control force of the chase spacecraft is illustrated in Figure 6, which always is limited within 0.2N. In addition, the trajectories of the encrypted relative position, relative velocity, and control input are shown in . Based on these figures, it is impossible for malicious to infer the actual relative motion information (Figs. 3 and 4) and the actual control command (Fig. 5) only by eavesdropping the encrypted signals (Figs. 7-9). Therefore, the proposed encrypted control framework not only achieves the desired relative motion of micro-spacecraft with graceful control performance but also ensures the secure information exchange among different components of the spacecraft.

Conclusions
This study proposed a novel encrypted control framework for spacecraft relative motion control using a logarithmic quantizer, two uniform quantizers, and a semi-homomorphic cryptosystem. The logarithmic quantizer was used to quantize the continuous relative state information, while the uniform quantizer was regarded as the encoder and decoder before encryption and after decryption, respectively. By selecting the proper quantization parameter, it is shown that the proposed encrypted control is capable of  guaranteeing ultimately uniformly bounded stability of the spacecraft relation motion system. Moreover, the security of the sensitive relative state information was ensured by the Paillier cryptosystem. Possible future work will concentrate on the encrypted control problem of spacecraft relative motion with more practical constraints, such as unknown parameter uncertainties, exogenous disturbances, and with a less conservative fully homomorphic encryption scheme.