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Derivative-based Optimization: Lie Algebra Method
Published in Kenichi Kanatani, 3D Rotations, 2020
This method has a big advantage over the parameterization, because any parameterization of rotation, such as axis-angle, Euler angles, and quaternions, has singularities at some special parameter values; they depend on parameterization but are likely to occur when R is the identity, a half-rotation, and a 360° rotation. At such singularities, the parameter values may have indeterminacy or their small changes may cause no rotations (except for high order terms). The gimbal lock of Euler angles described in Sec. 3.3 is a typical example. For numerical computation, numerical instability is likely to occur near such singularities. Theoretically, there exists no parameterization that does not have singularities, i.e., a smooth mapping onto the entire 3D space1, though encountering such singularities is very rare in practical applications. Using the Lie algebra method, however, one need not worry about any singularities at all, because all we do is to update the current rotation by adding a small rotation. In a sense, this is obvious, but not many people understand this fact.
Pushing Location Awareness to Context-Aware Augmented Reality
Published in Kaikai Liu, Xiaolin Li, Mobile SmartLife via Sensing, Localization, and Cloud Ecosystems, 2017
Rotation Matrix. Euler angles are simple and intuitive. On the other hand, Euler angles are limited by a phenomenon called “Gimbal lock.” Due to this reason, we use the DCM, a.k.a. the rotation matrix R $ \mathbf R $ , to compute the orientation and coordinate transformation. The elements of matrix R $ \mathbf R $ are the cosines of the unsigned angles between the body axes and the navigation axes.
Mass data methods
Published in W. Schofield, M. Breach, Engineering Surveying, 2007
Euler angles give a good representation of vehicle attitude and may be used for driving ship and aircraft cockpit displays including compass cards, which are about the yaw axis, and for aircraft artificial horizons, which are about the pitch and roll axes.
Event-triggered control for trajectory tracking of quadrotor unmanned aerial vehicle
Published in Systems Science & Control Engineering, 2022
The mathematical model of the QUAV position subsystem can be expressed as Wang et al. (2017)where and denote the position vector and velocity vector of the QUAV. is the input of the position subsystem, represents the sum of the lift generated by the four rotors. is the Euler angle vector, where ψ, θ, ϕ are the yaw, pitch and roll angle, which rotate around the Z, Y and X axes respectively. The nonlinear external disturbance is express as . Moreover, , m, g are the air damping coefficient, the mass of the QUAV and the acceleration of the gravity.
Numerical method for inverse kinematics using an extended angle-axis vector to avoid deadlock caused by joint limits*
Published in Advanced Robotics, 2021
Masanori Sekiguchi, Naoyuki Takesue
In planar robots, the domain of the orientation error can be easily changed because the orientation of the end effector can be represented by one variable. However, when the orientation of the end effector is expressed by three variables, such as Euler angles, the domain of the orientation error depends on the range of the inverse trigonometric function and cannot be changed easily. In addition to the above problems, Euler angles have the problem that gimbal lock and continuity of values are not guaranteed. To solve the deadlock problem, the authors have developed an extended angle-axis vector, which is a new method for expressing orientation errors [9]. In this paper, through application of the extended angle-axis vector, a new numerical inverse kinematics method is proposed for avoiding deadlock.
Attitude estimation of connected drones based on extended Kalman filter under real outdoor environments
Published in Advanced Robotics, 2022
Kento Fukuda, Shin Kawai, Hajime Nobuhara
In this section, we demonstrate the effectiveness of the proposed method through estimation simulations and outdoor experiments under various situations. We assume that the number of connections is two because attitude estimation is most difficult due to the number of sensors. We experiment with a situation where drones are connected side by side, as shown in Figure 2. We use the Euler angle for accuracy evaluation because it is more intuitive than quaternions. The Euler angle consists of three rotation angles, roll, pitch and yaw angles, which represent the rotation around x-, y- and z-axes, respectively. In this study, the World coordinate system rotates with respect to the Body coordinate system in the order of yaw, pitch and roll.