The precise and robust control of a quadrotor drone requires simultaneous management of its translational (position) and rotational (attitude) motion. Traditional control architectures often decouple or simply concatenate these six-degree-of-freedom (6-DoF) dynamics. While simpler, this approach treats the inherently coupled motions separately, potentially overlooking their interactions and complicating controller design within heterogeneous algebraic frameworks. To address this, the concept of integrated pose (position and attitude) modeling and control has gained prominence. The dual quaternion offers a unified eight-parameter representation for 6-DoF motion, enabling elegant, singularity-free modeling and facilitating integrated controller synthesis, as demonstrated in spacecraft and, more recently, quadrotor drone applications. However, the dual quaternion inherits a key limitation from its rotational quaternion part: it requires satisfaction of a normalization constraint $\hat{q}^*\hat{q} = 1$, where $\hat{q}$ is the unit dual quaternion and $*$ denotes conjugation. This constraint introduces parameter redundancy (eight parameters for six degrees of freedom) and necessitates additional computational steps to maintain, which can be problematic near singular numerical conditions. This paper addresses these issues by proposing a novel six-parameter representation based on the twistor, which provides a unified description for quadrotor drone pose without normalization constraints or parameter redundancy. Furthermore, to enhance the robustness and performance of the quadrotor drone system against disturbances and model uncertainties, a sliding mode controller is designed within this twistor framework.
1. Theoretical Foundation and Twistor-Based Modeling
We begin by establishing the mathematical groundwork. Let $\mathbb{R}^n$ denote the set of n-dimensional real vectors. A quaternion is defined as $q = q_0 + q_1 i + q_2 j + q_3 k = [q_0, \mathbf{q}_v^T]^T \in \mathbb{Q}$, where $q_0$ is the scalar part and $\mathbf{q}_v \in \mathbb{R}^3$ is the vector part. The operators $i, j, k$ are imaginary units satisfying $i^2=j^2=k^2=ijk=-1$. A dual number $\hat{a} = a_r + \epsilon a_d$ consists of a real part $a_r$ and a dual part $a_d$, with the dual unit $\epsilon$ satisfying $\epsilon^2=0$ but $\epsilon \neq 0$. A dual vector $\hat{\mathbf{a}} = \mathbf{a}_r + \epsilon \mathbf{a}_d$ belongs to $\mathbb{D}_v$. A dual quaternion $\hat{q} = q_r + \epsilon q_d$, where $q_r, q_d \in \mathbb{Q}$, belongs to $\mathbb{DQ}$. For a 3D vector $\mathbf{a}$, its quaternion form is $\underline{\mathbf{a}} = [0, \mathbf{a}^T]^T$.
The rigid-body pose (rotation and translation) from frame $O$ to frame $A$ can be represented by a unit dual quaternion $\hat{q}$. The rotation is described by the unit quaternion $q$, and the translation by the vector $\mathbf{r}_O$ expressed in frame $O$. The dual quaternion is constructed as:
$$
\hat{q} = q + \epsilon q_d, \quad \text{with} \quad q_d = \frac{1}{2} \underline{\mathbf{r}}_O q = \frac{1}{2} q \underline{\mathbf{r}}_A
$$
The kinematics are given by $\dot{\hat{q}} = \frac{1}{2} \hat{q} \hat{\underline{\Omega}}^O_{AO} = \frac{1}{2} \hat{\underline{\Omega}}^A_{AO} \hat{q}$, where $\hat{\Omega}^A_{AO} = \boldsymbol{\omega}^A_{AO} + \epsilon \mathbf{v}^A_{AO}$ is the dual velocity. Here, $\boldsymbol{\omega}$ is the angular velocity and $\mathbf{v} = \dot{\mathbf{r}} + \boldsymbol{\omega} \times \mathbf{r}$ is the translational velocity, fulfilling the motor transformation.
The Modified Rodrigues Parameters (MRP), a three-parameter representation of attitude, is defined as $\mathbf{p} = \frac{\langle \underline{q} \rangle_v}{1+q_0}$, where $\langle \cdot \rangle_v$ extracts the vector part. The MRP is singular at a rotation of $2\pi$ radians, but this is practically negligible as it corresponds to the identity rotation. Crucially, it has no normalization constraint. The twistor $\hat{B}$ is the dual extension of the MRP to represent full pose:
$$
\hat{B} = \frac{\hat{q} – 1}{\hat{q} + 1} \in \mathbb{DQ}_v
$$
This can be expanded into its dual vector form $\hat{\mathbf{B}} = \mathbf{p} + \epsilon \mathbf{b}$, where $\mathbf{b} = \boldsymbol{\Pi} \mathbf{r}_A$. The matrix $\boldsymbol{\Pi}$ is defined as:
$$
\boldsymbol{\Pi} = \frac{1}{4}(1 – \|\mathbf{p}\|^2)\mathbf{I}_3 – \frac{1}{2}[\mathbf{p} \times] + \frac{1}{2} \mathbf{p} \mathbf{p}^T
$$
Here, $[\mathbf{p} \times]$ is the skew-symmetric cross-product matrix of $\mathbf{p}$. The key properties of the twistor for quadrotor drone modeling are summarized and compared to dual quaternions below.
| Property | Dual Quaternion ($\hat{q}$) | Twistor ($\hat{\mathbf{B}}$) |
|---|---|---|
| Parameters | 8 (q_r, q_d) | 6 (p, b) |
| Constraint | Normalization: $\hat{q}^*\hat{q}=1$ | None |
| Singularity | None | At rotation of $2\pi$ (equivalent to identity) |
| Composition | $\hat{q}_{com} = \hat{q}_1 \otimes \hat{q}_2$ | $\hat{\mathbf{B}}_{com} = \frac{ (\hat{\mathbf{B}}_1 \oplus \hat{\mathbf{B}}_2) – (\hat{\mathbf{B}}_2 \oplus \hat{\mathbf{B}}_1)}{2} + \frac{\hat{\mathbf{B}}_1 + \hat{\mathbf{B}}_2}{1 – \hat{\mathbf{B}}_1 \cdot \hat{\mathbf{B}}_2}$ (Simplified form) |
The twistor kinematics are derived from its definition and the dual quaternion kinematics:
$$
\dot{\hat{\mathbf{B}}} = \frac{1}{4} (1 + \hat{\underline{B}}) \hat{\underline{\Omega}}^A_{AO} (1 – \hat{\underline{B}})
$$
Where $\hat{\underline{B}}$ is the dual quaternion form of the twistor. The transformation of dual velocity between frames is $\hat{\underline{\Omega}}^A_{AO} = \frac{1-\hat{\underline{B}}}{1+\hat{\underline{B}}} \hat{\underline{\Omega}}^O_{AO} \frac{1+\hat{\underline{B}}}{1-\hat{\underline{B}}}$.
2. Integrated Dynamics of a Quadrotor Drone in Twistor Form
We define three coordinate frames: the inertial frame $I$, the quadrotor drone body frame $B$, and a desired frame $D$. The relative pose of $B$ with respect to $D$ is described by the twistor $\hat{\mathbf{B}}_{BD}$. The relative kinematics are:
$$
\dot{\hat{\mathbf{B}}}_{BD} = \frac{1}{4} (1 + \hat{\underline{B}}_{BD}) \hat{\underline{\Omega}}^B_{BD} (1 – \hat{\underline{B}}_{BD})
$$
The relative dual velocity $\hat{\Omega}^B_{BD} = \hat{\Omega}^B_{BI} – \hat{\Omega}^B_{DI}$, where $\hat{\Omega}^B_{DI}$ is the desired dual velocity expressed in the $B$ frame via the appropriate twistor transformation.
The dynamics of a rigid quadrotor drone are derived using the dual momentum. We define the dual mass operator $\underline{\tilde{M}} = m \frac{d}{d\epsilon} + \epsilon \mathbf{J}$, where $m$ is the mass and $\mathbf{J}$ is the inertia matrix of the quadrotor drone. The action of this operator on a dual velocity $\hat{\Omega} = \boldsymbol{\omega} + \epsilon \mathbf{v}$ is $\underline{\tilde{M}} \hat{\underline{\Omega}} = m\mathbf{v} + \epsilon \mathbf{J}\boldsymbol{\omega}$. Let $\hat{\mathbf{F}} = \mathbf{f} + \epsilon \boldsymbol{\tau}$ be the dual force (force and torque) acting on the drone. Newton-Euler dynamics in dual form are:
$$
\underline{\tilde{M}} \, \dot{\hat{\underline{\Omega}}}^B_{BI} + \hat{\underline{\Omega}}^B_{BI} \times (\underline{\tilde{M}} \, \hat{\underline{\Omega}}^B_{BI}) = \hat{\underline{F}}^B
$$
Expanding this into its real (translational) and dual (rotational) parts yields the standard Newton and Euler equations for a quadrotor drone, confirming the model’s correctness:
$$
\begin{aligned}
&\text{Real Part:} \quad m \dot{\mathbf{v}}^B + m \boldsymbol{\omega}^B \times \mathbf{v}^B = \mathbf{f}^B \\
&\text{Dual Part:} \quad \mathbf{J} \dot{\boldsymbol{\omega}}^B + \boldsymbol{\omega}^B \times \mathbf{J} \boldsymbol{\omega}^B = \boldsymbol{\tau}^B
\end{aligned}
$$
For control design, we consider the forces and moments acting on the quadrotor drone: gravity $\hat{\mathbf{F}}^B_g$, control inputs from the rotors $\hat{\mathbf{F}}^B_u$, and disturbances $\hat{\mathbf{F}}^B_d$. The control force $\mathbf{f}_u$ and torque $\boldsymbol{\tau}_u$ are related to the rotor speeds $\omega_i$ ($i=1,\dots,4$) by:
$$
\begin{aligned}
f_u &= c_T (\omega_1^2 + \omega_2^2 + \omega_3^2 + \omega_4^2) \\
\tau_{ux} &= c_T d \left( \frac{\sqrt{2}}{2}\omega_1^2 – \frac{\sqrt{2}}{2}\omega_2^2 – \frac{\sqrt{2}}{2}\omega_3^2 + \frac{\sqrt{2}}{2}\omega_4^2 \right) \\
\tau_{uy} &= c_T d \left( \frac{\sqrt{2}}{2}\omega_1^2 + \frac{\sqrt{2}}{2}\omega_2^2 – \frac{\sqrt{2}}{2}\omega_3^2 – \frac{\sqrt{2}}{2}\omega_4^2 \right) \\
\tau_{uz} &= c_M (\omega_1^2 – \omega_2^2 + \omega_3^2 – \omega_4^2)
\end{aligned}
$$
where $c_T$, $c_M$ are aerodynamic coefficients and $d$ is the arm length. The complete integrated relative dynamics for the quadrotor drone in twistor form are:
$$
\dot{\hat{\underline{\Omega}}}^B_{BD} = \underline{\tilde{M}}^{-1} \hat{\underline{F}}^B_u + \underline{\tilde{M}}^{-1} (\hat{\underline{F}}^B_g + \hat{\underline{F}}^B_d) – \underline{\tilde{M}}^{-1} [\hat{\underline{\Omega}}^B_{BI} \times (\underline{\tilde{M}} \hat{\underline{\Omega}}^B_{BI})] – \dot{\hat{\underline{\Omega}}}^B_{DI}
$$
Equations (1) and (2) together form the integrated pose (position and attitude) error model for the quadrotor drone, suitable for controller design.
3. Sliding Mode Controller Design within the Twistor Framework
To achieve robust tracking for the quadrotor drone, a sliding mode controller (SMC) is designed. The SMC is renowned for its robustness against matched disturbances and parameter variations, which are common in quadrotor drone flight. The design is inspired by the MRP-based attitude SMC and extended to the 6-DoF twistor case.
We first define a dual sliding surface $\hat{\mathbf{s}}$ that incorporates both pose and velocity errors:
$$
\hat{\underline{s}} = \hat{\underline{\Omega}}^B_{BD} + \tilde{K}_1 \tilde{E} \left\langle \frac{4}{1 – \hat{\underline{B}}_{BD}^2} \hat{\underline{B}}_{BD} \right\rangle_v
$$
Here, $\tilde{K}_1 = k_1 \frac{d}{d\epsilon} + k_1 \epsilon$ is a dual gain operator with $k_1 > 0$, and $\tilde{E} = \mathbf{I}_3 \frac{d}{d\epsilon} + \mathbf{I}_3 \epsilon$ is the dual swap operator ($\tilde{E}(\mathbf{a}_r + \epsilon \mathbf{a}_d) = \mathbf{a}_d + \epsilon \mathbf{a}_r$). The notation $\langle \cdot \rangle_v$ extracts the vector part of the dual quaternion argument. In dual vector form, this surface decomposes into real (attitude/angular) and dual (position/linear) parts:
$$
\begin{aligned}
\mathbf{s}_r &= \boldsymbol{\omega}^B_{BD} + \frac{4k_1}{1+\|\mathbf{p}_{BD}\|^2} \mathbf{p}_{BD} \\
\mathbf{s}_d &= \mathbf{v}^B_{BD} + \frac{4k_1}{1+\|\mathbf{p}_{BD}\|^2} \mathbf{b}_{BD} – \frac{8k_1 (\dot{\mathbf{p}}_{BD} \cdot \mathbf{b}_{BD})}{(1+\|\mathbf{p}_{BD}\|^2)^2} \mathbf{p}_{BD}
\end{aligned}
$$
To drive the system to this sliding surface and mitigate chattering, a saturation function-based reaching law is chosen. Let $\text{sat}(\mathbf{x}, \sigma) = \mathbf{x}/\sigma$ if $\|\mathbf{x}\| < \sigma$, and $\text{sgn}(\mathbf{x})$ otherwise. The dual reaching law is:
$$
\dot{\hat{\underline{s}}} = -\tilde{K}_2 \, \text{sat}(\hat{\underline{s}}, \sigma)
$$
where $\tilde{K}_2 = k_2 \frac{d}{d\epsilon} + k_2 \epsilon$, $k_2 > 0$, and $\sigma > 0$ is the boundary layer thickness. By differentiating the sliding surface (3) and substituting the dynamics (2) and reaching law (5), the integrated control law for the quadrotor drone is derived:
$$
\begin{aligned}
\hat{\underline{F}}^B_u = &- \underline{\tilde{M}} \tilde{K}_2 \, \text{sat}(\hat{\underline{s}}, \sigma) + \hat{\underline{\Omega}}^B_{BI} \times (\underline{\tilde{M}} \hat{\underline{\Omega}}^B_{BI}) + \underline{\tilde{M}} \dot{\hat{\underline{\Omega}}}^B_{DI} \\
&- \underline{\tilde{M}} \tilde{K}_1 \tilde{E} \left\langle \frac{4}{(1 – \hat{\underline{B}}_{BD}^2)^2} (\dot{\hat{\underline{B}}}_{BD} + \hat{\underline{B}}_{BD} \dot{\hat{\underline{B}}}_{BD} \hat{\underline{B}}_{BD}) \right\rangle_v – \hat{\underline{F}}^B_g – \hat{\underline{F}}^B_d
\end{aligned}
$$
4. Stability Analysis via Lyapunov Theory
The stability of the closed-loop quadrotor drone system under the proposed twistor-based SMC is proven using Lyapunov’s direct method and LaSalle’s invariance principle. We analyze the reaching phase and the sliding phase separately.
Reaching Phase: Consider the Lyapunov function candidate $V_1 = \frac{1}{2} [\hat{\mathbf{s}} | \tilde{E} \hat{\mathbf{s}}]$, where the operator $[\hat{\mathbf{a}} | \hat{\mathbf{b}}] = \mathbf{a}_r \cdot \mathbf{b}_d + \mathbf{a}_d \cdot \mathbf{b}_r$. This is equivalent to $V_1 = \frac{1}{2}(\|\mathbf{s}_r\|^2 + \|\mathbf{s}_d\|^2) \geq 0$, which is zero only when $\hat{\mathbf{s}} = 0$. Its time derivative along the system trajectories, using the reaching law (5), is:
$$
\dot{V}_1 = [\dot{\hat{\mathbf{s}}} | \tilde{E} \hat{\mathbf{s}}] = -k_2 [\text{sat}(\hat{\mathbf{s}}, \sigma) | \tilde{E} \hat{\mathbf{s}}] = -k_2 \left( \text{sat}(\mathbf{s}_r, \sigma) \cdot \mathbf{s}_r + \text{sat}(\mathbf{s}_d, \sigma) \cdot \mathbf{s}_d \right) \leq 0
$$
Since $\dot{V}_1$ is negative definite with respect to $\hat{\mathbf{s}}$, the system state is driven to the boundary layer $\|\hat{\mathbf{s}}\| < \sigma$ in finite time. Inside the boundary layer, the dynamics become continuous, and the system converges to the sliding manifold $\hat{\mathbf{s}} = 0$ asymptotically.
Sliding Phase: On the sliding manifold $\hat{\mathbf{s}}=0$, the closed-loop dynamics reduce to $\hat{\Omega}^B_{BD} = -\tilde{K}_1 \tilde{E} \langle \frac{4}{1 – \hat{B}_{BD}^2} \hat{B}_{BD} \rangle_v$. Consider the Lyapunov function $V_2 = \frac{1}{2} [\hat{\mathbf{B}}_{BD} | \tilde{E} \hat{\mathbf{B}}_{BD}] = \frac{1}{2}(\|\mathbf{p}_{BD}\|^2 + \|\mathbf{b}_{BD}\|^2) \geq 0$. Its derivative on the sliding manifold becomes:
$$
\dot{V}_2 = [\dot{\hat{\mathbf{B}}}_{BD} | \tilde{E} \hat{\mathbf{B}}_{BD}] = -k_1 [\hat{\mathbf{B}}_{BD} | \tilde{E} \hat{\mathbf{B}}_{BD}] = -k_1 (\|\mathbf{p}_{BD}\|^2 + \|\mathbf{b}_{BD}\|^2) \leq 0
$$
By LaSalle’s invariance principle, the system converges to the largest invariant set where $\dot{V}_2 = 0$, which implies $\hat{\mathbf{B}}_{BD} = 0$ and consequently $\hat{\Omega}^B_{BD} = 0$. Therefore, the origin $(\hat{\mathbf{B}}_{BD}, \hat{\Omega}^B_{BD}) = (0, 0)$ is globally asymptotically stable. This proves that the quadrotor drone’s pose and velocity errors converge to zero under the proposed controller.
5. Simulation Results and Performance Analysis
The proposed twistor-based model and controller were validated through numerical simulations. The quadrotor drone parameters were: mass $m = 1.2$ kg, inertia $\mathbf{J} = \text{diag}(0.125, 0.125, 0.25)$ kg·m². Initial pose errors were set, and the controller gains were tuned to $k_1=3$, $k_2=20$, with a boundary layer $\sigma=0.9$. Two disturbance scenarios were considered to test robustness.
Scenario 1 (Moderate Disturbance):
$\mathbf{f}^B_d = [0.02\sin(0.6t), 0.01\cos(0.8t), 0.03\sin(0.3t)]^T$ N
$\boldsymbol{\tau}^B_d = [0.005\sin(0.2t), 0.001\cos(0.1t), 0.005\sin(0.3t)]^T$ N·m
Scenario 2 (Stronger Disturbance):
$\mathbf{f}^B_d = [0.5\sin(0.6t), 0.4\cos(0.8t), 0.1\sin(0.3t)]^T$ N
$\boldsymbol{\tau}^B_d = [0.1\sin(0.2t), 0.1\cos(0.1t), 0.1\sin(0.3t)]^T$ N·m
First, the equivalence of the twistor model to a standard MRP/vector model was verified in an open-loop simulation. The difference between the two models’ states was negligible, with mean squared errors essentially zero, confirming the correctness of the twistor-based quadrotor drone dynamics.
For closed-loop performance, the proposed Twistor-based Sliding Mode (TSM) controller was compared against a conventional Twistor-based PD controller. The PD control law was: $\hat{\mathbf{F}}^B_u = -4\tilde{K}_p \hat{\mathbf{B}}_{BD} – \tilde{K}_d \hat{\Omega}^B_{BD} + \text{dynamics compensation terms}$, with $k_p=20$, $k_d=10$. The performance metrics are summarized below.
| Scenario | Controller | Mean Position Error (m) | Mean Attitude Error (Quaternion Norm) | Settling Time (s, approx.) |
|---|---|---|---|---|
| 1 (Moderate) | TSM (Proposed) | 0.3833 | 0.0030 | 1.43 |
| PD | 0.6867 | 0.0076 | 2.44 | |
| 2 (Strong) | TSM (Proposed) | 0.3813 | 0.0031 | 1.73 |
| PD | 0.7093 | 0.0078 | 2.97 |
The simulation results demonstrate the effectiveness of the twistor-based integrated modeling. The quadrotor drone’s position and attitude errors converged to a small neighborhood of zero rapidly. The key advantages of the proposed TSM controller are clearly evident:
- Superior Robustness: Under both disturbance scenarios, the TSM controller maintained significantly lower steady-state errors in both position and attitude compared to the PD controller. The error for the PD controller was roughly double that of the TSM controller.
- Faster Convergence: The settling time for the TSM controller was consistently shorter, highlighting its quicker dynamic response.
- Effective Chatter Suppression: The use of the saturation function in the reaching law successfully mitigated the chattering phenomenon typical of SMC, resulting in smooth control signals suitable for the quadrotor drone’s actuators.
6. Conclusion
This paper has presented a novel framework for the integrated modeling and control of a quadrotor drone’s position and attitude using twistor theory. The twistor, a six-parameter representation derived from the dual extension of the Modified Rodrigues Parameters, successfully unifies the description of translational and rotational motion. It eliminates the normalization constraint and parameter redundancy associated with dual quaternions, offering a more efficient and straightforward algebraic foundation for modeling the quadrotor drone’s 6-DoF dynamics. Based on this model, an integrated sliding mode controller was designed within the twistor framework. The controller’s global asymptotic stability was rigorously proven using Lyapunov theory and LaSalle’s principle. Comprehensive simulations validated the accuracy of the twistor-based quadrotor drone model and demonstrated the superior performance of the proposed controller. Compared to a standard PD controller, the twistor-based SMC exhibited significantly enhanced robustness against external disturbances, faster convergence, and effective chattering suppression, making it a compelling approach for high-performance, robust flight control of quadrotor drones in demanding environments. Future work will focus on experimental validation and extending the controller to manage more complex scenarios such as aggressive maneuvering and payload transportation for the quadrotor drone.