Exponential Coordinates Research Articles

Attitude tracking of rigid bodies using exponential coordinates and a disturbance observer

In this paper, we study the problem of attitude trajectory tracking of rigid bodies subject to external disturbances. The attitude control problem has been studied in the last decades and novel solutions have been proposed. Nevertheless, most of the proposed controllers only achieve local or almost global asymptotic stability without considering external disturbances. Contrary to other works, we focus on the problem of achieving almost global exponential stability of the attitude tracking errors in the presence of external disturbances using continuous control laws. We propose an attitude trajectory tracking controller based on the exponential coordinates of rotation in combination with a disturbance observer that estimates the exogenous signals. To solve this problem, we adopt a hierarchical approach, where the proposed control law is divided into a kinematic controller (outer control loop) and a velocity tracking controller (inner control loop). To design the disturbance observer, it is assumed that exogenous signals can be generated by a linear exosystem, i.e. the magnitude and phase of the disturbance are unknown. The almost global exponential stability of the closed-loop dynamics' equilibrium point was proved by a strict Lyapunov function. The performance of the proposed approach is assessed by numerical simulations and experimental tests on a low-cost quadrotor.

On the Construction of Confidence Regions for Uncertain Planar Displacements.

This paper studies the statistical concept of confidence region for a set of uncertain planar displacements with a certain level of confidence or probabilities. Three different representations of planar displacements are compared in this context and it is shown that the most commonly used representation based on the coordinates of the moving frame is the least effective. The other two methods, namely the exponential coordinates and planar quaternions, are equally effective in capturing the group structure of SE(2). However, the former relies on the exponential map to parameterize an element of SE(2), while the latter uses a quadratic map, which is often more advantageous computationally. This paper focus on the use of planar quaternions to develop a method for computing the confidence region for a given set of uncertain planar displacements. Principal component analysis (PCA) is another tool used in our study to capture the dominant direction of movements. To demonstrate the effectiveness of our approach, we compare it to an existing method called rotational and translational confidence limit (RTCL). Our examples show that the planar quaternion formulation leads to a swept volume that is more compact and more effective than the RTCL method, especially in cases when off-axis rotation is present.

Efficient Kinematic Calibration for Articulated Robot Based on Unit Dual Quaternion

Removing the parameter redundancy in the kinematic error model does improve the robustness of parameter identification. However, this does not help much on saving the calculation time on the error compensation process, which is closely related to the data structure used for representing rigid-body motion. Compared with the homogenous transformation matrix (HTM), the unit dual quaternion (UDQ) can describe the rigid-body motion in term of an array with eight entries, thus it is a conducive data structure for efficient kinematic computation. In this work, the exponential and logarithm mappings of a UDQ are defined explicitly, so that the relationship between finite and instantaneous motions is set up under the exponential coordinate. Thereafter, by defining the adjoint operator of UDQ to transform the twist under different frames, the linearized kinematic error models are built with local product-of-exponentials (POE) formula. This facilitates the upcoming parameter identification and error compensation processes. Specially, the counts of elementary operations are evaluated throughout the compensation processes, which shows the number of arithmetic operations are greatly reduced compared with using POE formula based on HTM. Additionally, the simple data structure in term of 1-D arrays saves the data dispatch time when performing adjoint operations. These make the developed model suitable for calibration of articulated robot with a long sequence of trajectories. Experiments on a 6-degree-of-freedom industrial robot validate the effectiveness of proposed approach.

Highly accurate differentiation of the exponential map and its tangent operator

Exponential coordinates are widely used in simulation codes for flexible multibody systems based on a Lie group approach. The accurate and efficient evaluation of the exponential map, the tangent operator and its higher order derivatives is crucial. This paper presents a systematic derivation process based on the matrix series expansion of the exponential map and of the tangent operator. This approach is general as it can be applied to any matrix Lie group. For the Lie groups SO(3) and SE(3), the closed form expression of these operators can also be established and is summarized in the paper. It is shown that the closed form of the operators is affected by round-off errors for small rotation amplitudes, whereas the series form is affected by truncation errors at high rotation amplitudes. The computational efficiency of the two approaches is also discussed. A switching strategy between the closed form and the series form is then proposed to obtain an adjustable compromise between accuracy and computational cost.

On the sensitivity of pose estimation neural networks: Rotation parameterizations, Lipschitz constants, and provable bounds

In this paper, we approach the task of determining sensitivity bounds for pose estimation neural networks. This task is particularly challenging as it requires characterizing the sensitivity of 3D rotations. We develop a sensitivity measure that describes the maximum rotational change in a network’s output with respect to a Euclidean change in its input. We show that this measure is a type of Lipschitz constant, and that it is bounded by the product of a network’s Euclidean Lipschitz constant and an intrinsic property of a rotation parameterization which we call the “distance constant”. We derive the distance constant for several rotation parameterizations, and then discuss why the structure of most of these parameterizations makes it difficult to construct a pose estimation network with provable sensitivity bounds. However, we show that sensitivity bounds can be computed for networks which parameterize rotation using unconstrained exponential coordinates. We then construct and train such a network and compute sensitivity bounds for it.

Rigid-Body Noncertainty Equivalence Adaptive Control on the Tangent Bundle of SE(3)

This paper addresses an adaptive rigid-body pose-tracking control problem with unknown system parameters. The proposed control is based on a noncertainty equivalence (NCE) adaptive principle applied to the configuration manifold of rigid-body motion, i.e., special Euclidean group SE(3), and its tangent bundle TSE(3). The controller drives the system to follow an arbitrary reference trajectory with bounded time derivatives. The system states are composed of the elements of SE(3), consisting of rotations and translations, and the six-dimensional vector of angular and translational velocities in Euclidean space. Almost-global-asymptotic stability of the system is demonstrated using a Morse–Lyapunov stability analysis. Then, the performance of the proposed controller is verified and is compared to the results obtained by a certainty-equivalence (CE) adaptive control applied to exponential coordinates associated with the linear space of the Lie algebra . It is shown that the TSE(3)-based NCE adaptive control design can eliminate the performance degradation due to the cancellation of uncertain parameter effects that would otherwise appear when the CE adaptive controller is used. As a result, the proposed NCE-based adaptive pose-tracking controller recovers the closed-loop pose-tracking controller performance with the system parameters fully known to the controller.

Attitude Synchronization of a Group of Rigid Bodies Using Exponential Coordinates.

Currently, managing a group of satellites or robot manipulators requires coordinating their motion and work in a cooperative way to complete complex tasks. The attitude motion coordination and synchronization problems are challenging since attitude motion evolves in non-Euclidean spaces. Moreover, the equation of motions of the rigid body are highly nonlinear. This paper studies the attitude synchronization problem of a group of fully actuated rigid bodies over a directed communication topology. To design the synchronization control law, we exploit the cascade structure of the rigid body's kinematic and dynamic models. First, we propose a kinematic control law that induces attitude synchronization. As a second step, an angular velocity-tracking control law is designed for the dynamic subsystem. We use the exponential coordinates of rotation to describe the body's attitude. Such coordinates are a natural and minimal parametrization of rotation matrices which almost describe every rotation on the Special Orthogonal group SO(3). We provide simulation results to show the performance of the proposed synchronization controller.

Adaptive fuzzy dynamic surface fault-tolerant control for coupled spacecraft with actuator faults and saturation

This study investigates the fault-tolerant control problem associated with pose tracking for coupled spacecraft in the presence of both actuator faults and actuator saturation constraint. By using the Lie group SE(3), a six-degrees of freedom (6-DOF) coupled kinematics and dynamics model is established to derive the relative motion tracking errors in the exponential coordinates. Then, all system states are reconstructed to develop a dynamic surface fault-tolerant controller. Considering the lumped disturbance consisting of internal uncertainties, external unknown disturbances and actuator faults, a fuzzy adaptive method is used to estimate and compensate for the lumped disturbance. Next, the controller is nested with an auxiliary first-order filter to compensate for the saturation constraints. A Lyapunov analysis is used to prove the stability and finite time convergence of the proposed controller. Finally, The tracking performance is verified through a numerical example.

Logarithmic catastrophes and Stokes’s phenomenon in waves at horizons

Waves propagating near an event horizon display interesting features including logarithmic phase singularities and caustics. We consider an acoustic horizon in a flowing Bose–Einstein condensate where the elementary excitations obey the Bogoliubov dispersion relation. In the Hamiltonian ray theory the solutions undergo a broken pitchfork bifurcation near the horizon and one might therefore expect the associated wave structure to be given by a Pearcey function, this being the universal wave function that dresses catastrophes with two control parameters. However, the wave function is in fact an Airy-type function supplemented by a logarithmic phase term, a novel type of wave catastrophe. Similar wave functions arise in aeroacoustic flows from jet engines, path integrals in radio astronomy, and also gravitational horizons if dispersion which violates Lorentz symmetry in the UV is included. The approach we take differs from most previous authors in that we analyze the behavior of the integral representation of the wave function using exponential coordinates. This allows for a different treatment of the branch cuts and gives rise to an analysis based purely on saddlepoint expansions. We are thereby able to resolve the multiple real and complex waves that interact at the horizon and its companion caustic. We find that the horizon is a physical manifestation of a Stokes surface, marking the place where a wave is born, and that the horizon and the caustic do not in general coincide: the finite spatial region between them delineates a broadened horizon.

Modeling and movement control of inverted two-wheeled robots using exponential coordinates

Modeling and movement control of inverted two-wheeled robots using exponential coordinates

Adaptive Control on SE(3) for Spacecraft Pose Tracking With Harmonic Disturbance and Input Saturation

This article investigates the coupled attitude and orbit tracking maneuver control problem of a spacecraft system in close space missions with model uncertainty, unknown disturbance, and input saturation. An adaptive compact control scheme with featuring of disturbance rejection and antiwindup is reported. First, the exponential coordinates on Lie group SE(3) are employed to describe the relative pose, which includes the relative position and attitude, and the tracking error system is established by introducing an instrumental error variable, which contains exponential coordinates and velocity tracking error. Then, a dynamic disturbance compensator is constructed based on the idea of internal model design to finely compensate for a class of unknown harmonic disturbances. Furthermore, a nonsingular auxiliary system is proposed to overcome input saturation, in which a time-varying parameter is introduced to improve the performance of the antiwindup auxiliary system. This parameter makes the closed-loop system possess two properties: the timeliness of saturation compensation and the asymptotic convergence of tracking errors. With the developed control law, the stability of the obtained feedback system is strictly proved by means of Lyapunov stability theory. Finally, the numerical simulation is carried out for an on-orbit servicing mission, which validates the effectiveness of the proposed integrated controller in solving the pose tracking control problem.

Emergent Behaviors of Rotation Matrix Flocks

We derive an explicit form for the Cucker-Smale (CS) model on the special orthogonal group $\mathrm{SO}(3)$ by identifying closed form expressions for geometric quantities such as covariant derivative and parallel transport in exponential coordinates. We study the emergent dynamics of the model by using a Lyapunov functional approach and La Salle's invariance principle. Specifically, we show that velocity alignment emerges from some admissible class of initial data, under suitable assumptions on the communication weight function. We characterize the $\omega$-limit set of the dynamical system and identify a dichotomy in the asymptotic behavior of solutions. Several numerical examples are provided to support the analytical results.

Consensus Control of Rigid Body Spacecraft in Orbital Relative Motion using TSE(3) and Exponential Coordinates

Consensus Control of Rigid Body Spacecraft in Orbital Relative Motion using TSE(3) and Exponential Coordinates

Robust Terminal Sliding Mode Control on SE(3) for Gough–Stewart Flight Simulator Motion Platform with Payload Uncertainty

This work proposes a robust terminal sliding mode control scheme on Lie group space SE(3) for Gough–Stewart flight simulator motion systems with payload uncertainty. A complete dynamic model with geometric mechanical structures and a computer dynamic model built in the MATLAB/Simulink package are briefly presented. The robust control strategy on the Lie group SE(3) is applied at the workspace level to counteract the effects of imperfect compensation due to model simplification and payload uncertainty in flight simulator application. With exponential coordinates for configuration error and adjoint operator on Lie algebra se(3), the robust control strategy is designed to guarantee almost global finite-time convergence over state space through the Lyapunov stability theory. Finally, a describing function and a step acceleration response to characterize the performance of a flight simulator motion base are employed to compare the robustness performance of the proposed controller on SE(3) with the conventional terminal sliding mode controller on Cartesian space. The comparison experimental results verify that the proposed controller on SE(3) provides better robustness than the conventional controller on Cartesian space, which means higher bandwidth in two degrees of freedom and faster response with smaller tracking error in six degrees of freedom.

Attitude Control of a Quadrotor with Fuzzy Logic Controller on SO(3)

A quadrotor is an unmanned aerial vehicle (UAV) with two pairs of rotors rotating in opposite directions. Some of its unique abilities are hovering and vertical take-off and landing (VTOL). Most recent works carried out the UAVs' rotation parametrization using Euler angles and a quaternion. Those UAVs suffer from singularities and ambiguities. A geometric control is generally used to deal with those problems. Exponential coordinate in the geometric control maps R3 into SO(3). This paper presented a fuzzy logic controller on SO(3) to control the attitude of the quadrotor. The input of the fuzzy logic controller is the angular velocity (ω) and exponential coordinate error of rotation (ζ), while the output is torque (τ). The error function in this controller is a rotation matrix on SO(3). This proposed controller can control the attitude of the quadrotor based on the expected attitude for maneuvers both on one axis and all axis with a steady-state error of about 0.02 rad.

Prescribed-time time-varying sliding mode based integrated translation and rotation control for spacecraft formation flying

<p style='text-indent:20px;'>This paper studies the prescribed-time spacecraft formation flying problem. The coupled 6-degrees-of-freedom kinematics and dynamics for spacecraft are modeled on Lie group SE(3), where the gravity-gradient torque, the perturbing forces caused by Earth's oblateness and unknown external disturbances are all taken into account. The relative configuration is expressed by using exponential coordinates of SE(3). In order to regulate the relative motion between the leader spacecraft and follower spacecraft, a novel prescribed-time control law is given based on a prescribed-time time-varying high-gain sliding mode method. Numerical simulations verify the effectiveness of the proposed control scheme.</p>

An Analytical Method for Decoupled Local Smoothing of Linear Paths in Industrial Robots

The linear-format path is widely adopted to approximate the continuous contour in robot controllers. The tangential discontinuity of the linear paths usually causes the discontinuity of the joint velocity. To comply with the joint kinematics limits, the robots have to stop at each corner point, resulting in a great loss of efficiency. To achieve a smooth motion, this paper presents an analytical decoupled C3 continuous local path smoothing method for industrial robots. The tool position path is smoothed in the reference frame while the tool orientation is smoothed in the rotation parametric space based on the exponential coordinates of rotations. The quintic B-splines are inserted at the corners of the linear segments to achieve the G3 continuity of the tool position path and tool orientation path. The orientation smoothing error is constrained analytically. By reparameterization of the remaining linear segments using specially constructed B-splines, the C3 continuity of the tool position path and tool orientation path is achieved. Then, the synchronization of the tool orientation path and tool position path can be guaranteed by sharing the same curve parameter. Besides, to improve the smoothness of the angular motion on the remaining linear segments during parameter synchronization, the transition lengths of the inserted B-splines are optimized. The proposed local smoothing method guarantees that the generated smooth orientation path in the rotation space is invariant with the selection of the reference frame and the orientation of the tool frame, and ensures the jerk-continuous motion with a smoother angular motion on the remaining linear segments, which could improve the motion smoothness of the tool path and tracking accuracy. The effectiveness of the proposed method is validated through simulation and experiments.

Real-time recognition of human lower-limb locomotion based on exponential coordinates of relative rotations

Real-time recognition of human lower-limb locomotion based on exponential coordinates of relative rotations

Extended state observer–based output feedback control for spacecraft pose tracking with control input saturation

This article proposed an extended state observer (ESO)–based output feedback control scheme for rigid spacecraft pose tracking without velocity feedback, which accounts for inertial uncertainties, external disturbances, and control input constraints. In this research, the 6-DOF tracking error dynamics is described by the exponential coordinates on SE(3). A novel continuous finite-time ESO is proposed to estimate the velocity information and the compound disturbance, and the estimations are utilized in the control law design. The ESO ensures a finite-time uniform ultimately bounded stability of the observation states, which is proved utilizing the homogeneity method. A non-singular finite-time terminal sliding mode controller based on super-twisting technology is proposed, which would drive spacecraft tracking the desired states. The other two observer-based controllers are also proposed for comparison. The superiorities of the proposed control scheme are demonstrated by theory analyses and numerical simulations.

Genetic-algorithm-based global design optimization of tree-type robotic systems involving exponential coordinates

This paper proposes the simultaneous design optimization of the geometry (joint position and directions) and the topology (joint distribution and connection) of tree-type robotic systems based on an exponential coordinate system expression. Tree-type systems represent a versatile system expression of mechanical systems comprising multiple serial link chains branching from the root. Previously, we derived general closed-form formulas of the kinematics and dynamics of tree-type systems in exponential coordinates by introducing a connectivity matrix called the chain matrix. In this study, these results are first extended to floating base and closed-chain systems to enlarge the system framework. Next, the efficient coding of the system parameters by using a genetic algorithm (GA) is demonstrated. The closed-form formulas of the kinematics and dynamics allow for cost evaluations through numerical simulations with feedback control. Design examples of a robotic platform and a grasping/manipulation system illustrate the proposed optimization process.