Unit Dual Quaternion Research Articles

Unit dual quaternion‐based pose optimisation for visual runway observations

This study addresses the pose estimation problem of an aircraft runway using visual observations in a landing approach scenario. The authors utilised the fact that the geodetic coordinates of most runways are known precisely with highly visible markers. Thus, the runway observations can increase the level of situational awareness during the landing approach, providing additional redundancy of navigation and less reliance on global positioning system. A novel pose optimisation algorithm is proposed utilising unit dual quaternion for the runway corner observations obtained from a monocular camera. The estimated runway pose is further fused with an inertial navigation system in an extended Kalman filter. An open-source flight simulator is used to collect and process the visual and flight dataset during the landing approach, demonstrating reliable runway pose estimates and the improved inertial navigation solution.

Dual Quaternion-Based Powered Descent Guidance with State-Triggered Constraints

This paper presents a numerical algorithm for computing six-degree-of-freedom free-final-time powered descent guidance trajectories. The trajectory generation problem is formulated using a unit dual quaternion representation of the rigid body dynamics and several standard path constraints. Our formulation also includes a special line-of-sight constraint that is enforced only within a specified band of slant ranges relative to the landing site: a novel feature that is especially relevant to terrain and hazard relative navigation. We use the newly introduced state-triggered constraints to formulate these range constraints in a manner that is amenable to real-time implementations. The resulting nonconvex optimal control problem is solved iteratively as a sequence of convex second-order cone programs that locally approximate the nonconvex problem. Each second-order cone program is solved using a customizable interior point method solver. Also introduced are a scaling method and a new heuristic technique that guide the convergence process toward dynamic feasibility. To demonstrate the capabilities of our algorithm, two numerical case studies are presented. The first studies the effect of including a slant-range-triggered line-of-sight constraint on the resulting trajectories. The second study performs a Monte Carlo analysis to assess the algorithm’s robustness to initial conditions and real-time performance.

Unscented Dual Quaternion Particle Filter for SE(3) Estimation

We present a novel dual quaternion filter for recursive estimation of rigid body motions. Based on the sequential Monte Carlo scheme, particles are deployed on the manifold of unit dual quaternions. This allows non-parametric modeling of arbitrary distributions underlying on the SE(3) group. The proposal distribution for importance sampling is estimated particle-wise by a novel dual quaternion unscented Kalman filter (DQ-UKF). It is adapted to the manifold geometric structure and drives the prior particles towards high-likelihood regions on the manifold. The resultant unscented dual quaternion particle filter (U-DQPF) incorporates the most recently observed evidence, raising the particle efficiency considerably for nonlinear pose estimation tasks. Compared with ordinary particle filters and other parametric model-based dual quaternion filtering schemes, the proposed U-DQPF shows superior performance in nonlinear SE(3) estimation.

Rigid-Body Pose Hybrid Control Using Dual Quaternions: Global Asymptotic Stabilization and Robustness

A hybrid feedback control scheme is proposed for stabilization of rigid-body dynamics (position, orientation, and velocities) using unit dual quaternions, in which the dual quaternions and velocities are used for feedback. Specifically, both set-point stabilization and tracking control are addressed in this work. It is well known that rigid-body attitude control is subject to topological constraints, which often result in discontinuous control to avoid the unwinding phenomenon. In contrast, the hybrid scheme allows the controlled system to be robust in the presence of uncertainties, which would otherwise cause chattering about the point of discontinuous control while also ensuring acceptable closed-loop response characteristics. The stability of the closed-loop system is guaranteed through a Lyapunov analysis and the use of an invariance principle for hybrid systems. Simulation results for a rigid-body model are presented to illustrate the performance of the proposed hybrid dual-quaternion feedback control schemes.

Resolved-Acceleration Control of Serial Robotic Manipulators Using Unit Dual Quaternions

A new method for resolved-acceleration control of serial chain manipulators has been proposed which uses dual quaternion representation of screw-based motion variables, i.e. pose, velocity and acceleration. The corresponding coupled controller considers both translation and orientation errors simultaneously for trajectory tracking and utilizes spatial acceleration to compute the feedforward compensation term for feedback linearization. The proposed coupled control law was validated on a robotic arm along a pre-defined trajectory. The controller demonstrated an improved trajectory tracking performance as compared to the conventional decoupled resolved-acceleration controller which treats translation and orientation error separately.

Operation mode analysis of lower-mobility parallel mechanisms based on dual quaternions

This paper aims to provide an efficient way for analyzing operation modes and revealing corresponding motion characteristics of lower-mobility parallel mechanisms (LMPMs) using unit dual quaternions. Unit dual quaternions are first classified into 132 cases corresponding to 132 elementary operation modes based on the number of constant zero components. Meanwhile, the kinematic interpretation of these cases is presented in detail. Then a general method for analyzing operation modes and revealing motion characteristics of LMPMs is proposed according to unit dual quaternions and geometrical constraints. By this means, operation modes of LMPMs with complicated constraint conditions can also be analyzed, where a prime decomposition of the ideal corresponding to constraint equations in this condition is infeasible. Taken a 3-RSR LMPM and a 3-RPS LMPM as examples, the operation modes and motion characteristics can be obtained by the proposed approach. It is shown that the former LMPM has seven operation modes including two elementary operation modes and five extra operation modes. Under certain operation modes, the zero-torsion motion of the 3-RSR LMPM can not even be achieved. On the other hand, the latter has two operation modes in which the parasitic motion exists. To gain a deeper insight into the physical meaning of the operation modes, axodes are analyzed and drawn by mean of the velocity operator. It is demonstrated that the 3-RSR LMPM can realize an equal-diameter spherical pure rolling movement with variable diameters and the 3-RPS LMPM can achieve a rolling movement accompanied by sliding.

Relative position and attitude coordinated control based on unit dual quaternion

In traditional algorithms, relative pose controls of a leader–follower rigid body consider the position and attitude as two subsystems in designing control laws. Since relative pose of a leader–follower rigid body is coupled, having a pose coupling control is a smarter choice. For deficiencies in decentralized control of relative pose, this article establishes a coupled dynamic and kinematic model of a leader–follower rigid body based on the dual-number and dual-quaternion theory framework. Furthermore, based on the proposed model, the sliding mode control theory is used to conduct coordinated control for relative pose of a leader–follower rigid body. The coupled algorithm does not need to design two sets of control laws for relative pose of a leader–follower rigid body; therefore, the complexity of the control system is reduced. Simulation results show that this algorithm does not only demonstrate the coupled control law of the relative pose but also show good tracking control performance.

Iterative solution of Helmert transformation based on a unit dual quaternion

The rigid motion involving both rotation and translation in the 3D space can be simultaneously described by a unit dual quaternion. Considering this excellent property, the paper constructs the Helmert transformation (seven-parameter similarity transformation) model based on a unit dual quaternion and then presents a rigid iterative algorithm of Helmert transformation using a unit dual quaternion. Because of the singularity of the coefficient matrix of the normal equation, the nine parameter (including one scale factor and eight parameters of a dual quaternion) Helmert transformation model is reduced into five parameter (including one scale factor and four parameters of a unit quaternion which can represent the rotation matrix) Helmert transformation one. Besides, a good start estimate of parameter is required for the iterative algorithm, hence another algorithm employed to compute the initial value of parameter is put forward. The numerical experiments involving a case of small rotation angles i.e. geodetic coordinate transformation and a case of big rotation angles i.e. the registration of LIDAR points are studied. The results show the presented algorithms in this paper are correct and valid for the two cases, disregarding the rotation angles are big or small. And the accuracy of computed parameter is comparable to the classic Procrustes algorithm from Grafarend and Awange (J Geod 77:66–76, 2003), the orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015), and the algorithm from Wang et al. (J Photogramm Remote Sens 94:63–69, 2014).

Inverse Dynamics for Discrete Geometric Mechanics of Multibody Systems With Application to Direct Optimal Control

In this paper, we present efficient algorithms for computation of the residual of the constrained discrete Euler–Lagrange (DEL) equations of motion for tree structured, rigid multibody systems. In particular, we present new recursive formulas for computing partial derivatives of the kinetic energy. This enables us to solve the inverse dynamics problem of the discrete system with linear computational complexity. The resulting algorithms are easy to implement and can naturally be applied to a very broad class of multibody systems by imposing constraints on the coordinates by means of Lagrange multipliers. A comparison is made with an existing software package, which shows a drastic improvement in computational efficiency. Our interest in inverse dynamics is primarily to apply direct transcription optimal control methods to multibody systems. As an example application, we present a digital human motion planning problem, which we solve using the proposed method. Furthermore, we present detailed descriptions of several common joints, in particular singularity-free models of the spherical joint and the rigid body joint, using the Lie groups of unit quaternions and unit dual quaternions, respectively.

Cartan connection applied to dynamic calculation in robotics

A Cartan connection is an important mathematical object in differential geometry that generalizes, to an arbitrary Riemannian space, the concept of angular velocity (and twists) of a non-inertial reference frame. In fact, this is an easy way to calculate the angular velocities (and twists) in systems with multiple reference frames, like a robot or a complex mechanism. The concept can also be used in different representations of rotations and twists, like in the Lie groups of unit quaternions and unit dual quaternions. In this work, the Cartan connection is applied to the kinematic and dynamic calculations of systems with multiple reference frames that could represent a robotic serial chain or mechanism. Differently from previous works, the transformation between frames is represented as unit quaternions, which are known to be better mathematical representations from the numerical point of view. It is also generalized to this new quaternion representation, previous results like the extended Newton’s equation, the covariant derivative and the Cartan connection. An example of application is provided.

Computationally Efficient Inverse Dynamics of a Class of Six-DOF Parallel Robots: Dual Quaternion Approach

Computationally efficient inverse dynamics is crucial to the real-time application of parallel robots. This paper provides a computationally more efficient solution to the inverse dynamics of a class of six-DOF parallel robots based on the dual quaternion approach under the principle of virtual power. A unit dual quaternion is selected as the generalized coordinates of the system. The equations of motion are then constructed by the principle of virtual power. The dual quaternion constraints are eliminated by the null space formulation to obtain the inverse dynamic solution. It is revealed that in the new solution, the Jacobian matrices and the orthogonal complement matrix are all linear with respect to the generalized coordinates. Additionally, the positions, velocities and accelerations of all bodies are quadratic with respect to the generalized coordinates, velocities and accelerations. Such succinct expressions render the new solution computationally more efficient. The execution time of the dual quaternion approach and the traditional one are compared under the same condition by two different six-DOF parallel robots: 6-UPS and 6-PUS. The results show that the new solution can save the computational cost by 43.45% and 38.45% respectively for the two robots, illustrating the effectiveness of the proposed approach.

A dual quaternion algorithm of the Helmert transformation problem

Rigid transformation including rotation and translation can be elegantly represented by a unit dual quaternion. Thus, a non-differential model of the Helmert transformation (3D seven-parameter similarity transformation) is established based on unit dual quaternion. This paper presents a rigid iterative algorithm of the Helmert transformation using dual quaternion. One small rotation angle Helmert transformation (actual case) and one big rotation angle Helmert transformation (simulative case) are studied. The investigation indicates the presented dual quaternion algorithm (QDA) has an excellent or fast convergence property. If an accurate initial value of scale is provided, e.g., by the solutions no. 2 and 3 of Závoti and Kalmár (Acta Geod Geophys 51:245–256, 2016) in the case that the weights are identical, QDA needs one iteration to obtain the correct result of transformation parameters; in other words, it can be regarded as an analytical algorithm. For other situations, QDA requires two iterations to recover the transformation parameters no matter how big the rotation angles are and how biased the initial value of scale is. Additionally, QDA is capable to deal with point-wise weight transformation which is more rational than those algorithms which simply take identical weights into account or do not consider the weight difference among control points. From the perspective of transformation accuracy, QDA is comparable to the classic Procrustes algorithm (Grafarend and Awange in J Geod 77:66–76, 2003) and orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015. https://doi.org/10.1186/s40623-015-0263-6).

A Method of Robot Base Frame Calibration by Using Dual Quaternion Algebra

When the robot is required to execute a certain task in the world coordinate system (WCS), it is necessary to find the coordinate transformation between the robot base coordinate system (RBCS) and WCS to enable the high precision motion planning. This paper presents a simple and accurate method that allows a simultaneous computation of the coordinate transformations (i.e., rotation and translation) from WCS to RBCS. Based on the dual quaternion, the robot kinematic model and formulas for calculating the transformation are derived, which allow calculating the rotation and translation simultaneously. Taking the unit dual quaternion as a constraint, the Lagrangian multiplier method is employed to obtain the optimum transformation. Both simulation and experiment results show that higher calibration precision is obtained. The proposed approach has certain reference value and guiding sense for other calibration problems.

Pose Estimation using Dual Quaternions and Moving Horizon Estimation

This paper presents a moving horizon estimator (MHE) for estimating pose (attitude and position) of a dynamic system where pose measurements are available in the form of unit dual quaternions. A unit dual quaternion is an 8 parameter nonsingular representation of pose and has previously been used for pose estimation with Kalman filters (KF). We formulate a cost function in terms of the quaternion product and propose a MHE that includes the N latest measurements in the estimation. In addition, we suggest a measurement relation based on the Cayley transform of the noise, where the noise has a Gaussian distribution about the x-y-z and roll-pitch-yaw parameters of the pose. The MHE is compared against the dual quaternion multiplicative extended KF (DQ-MEKF) and the twistor-based unscented KF (T-UKF) through 100 Monte Carlo simulations, where the simulated data is generated according to the defined system dynamics. It is found that the MHE gives more accurate pose estimation results.

Motion mapping of the heterogeneous master–slave system for intuitive telemanipulation

Heterogeneous master–slave robots are widely used as both the assistive robot system for the elder/disabled people and the teleoperation robot system in dangerous environments. For the sake of intuitive teleoperation, motion mapping should be applied to keep the slave arm possessing the similar configuration of the master arm in the whole course of telemanipulation. We propose a motion mapping on the joint analog level based on unit dual quaternions (UDQs) for the telecontrol of a heterogeneous slave robot arm via the motion of human master arm. First, we analogize the links of the slave robot to the links of the human master arm, and accordingly group the slave arm joints into “shoulder,” “elbow,” and “wrist” joint analog sets. Then, we capture the motion of the human master arm via a wearable motion capture system and take it as the motion order of the slave arm. Finally, we solve the motion of slave robot arm’s joint via UDQ-based 1-, 2-, and 3-Degree of freedom (DOF) orientation error UDQ decomposition algorithms that are suit for solving a typical class of robotic slave arms. As a result, the operator can manipulate the heterogeneous robot system intuitively and conveniently, and the fatigue degree and error rate of the user can be significantly reduced, and thus the safety of teleoperation in an unstructured and constrained environment can be significantly improved.

A novel approach to Lie group structured configuration spaces of rigid bodies

AbstractIn rigid body mechanics, a body's configuration consisting of orientation and position have to be described, as well as angular velocity and velocity. There are several different ways to do this, leading to different configuration spaces. Popular choices include SO(3) × ℝ3, SE(3) or unit dual quaternions. All these configuration spaces possess a Lie group structure. We present the novel approach 𝕊 ⋉ ℝ3, which is a configuration space similar to SE(3), but describes the orientation in terms of unit quaternions instead of rotation matrices. Furthermore we show its relation to and advantages over other configuration spaces. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Distributed attitude and translation consensus for networked rigid bodies based on unit dual quaternion

SummaryThis paper provides unified solutions for distributed attitude and translation consensus problems for networked fully actuated rigid bodies under the fixed and undirected communication topology with the tool of unit dual quaternion. We investigate two kinds of consensus, that is, leaderless consensus and leader‐following consensus with a static leader. Firstly, the dynamics of rigid bodies are presented by unit dual quaternion. The control inputs of rigid bodies are also obtained from unit dual quaternion. Secondly, we propose a distributed consensus law in the form of dual quaternion to guarantee that the attitudes and translations of all rigid bodies reach consensus, respectively, without a leader. Thirdly, the leader‐following consensus problem with a static leader is studied. With the proposed leader‐following consensus law, the states of all rigid bodies converge to the corresponding states of the static leader, including the attitude and the translation. Finally, numerical examples are provided to validate the effectiveness of the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

A dual quaternion solution to the forward kinematics of a class of six-DOF parallel robots with full or reductant actuation

The forward kinematics is the basis of the design and control of the parallel robots. This paper aims to provide an efficient solution to the forward kinematics of a class of six-degrees-of-freedom parallel robots for real-time applications. With a unit dual quaternion used as the generalized coordinates of the robot system, the forward kinematic equations are derived to be a set of quadratic ones. An efficient algorithm is proposed to get the actual solution to them. The convergence and singularity problems of the new algorithm have been discussed. We have provided a convergence strategy and revealed the internal relation of the singularity with that of the parallel robot, proving the feasibility of the algorithm and giving the working condition in the practical applications. The new algorithm have been compared to the Newton's method for an 8-UPS parallel robot, resulting in the time consumptions of 0.2187 milliseconds and 14.25 milliseconds respectively. And then we perform a simulation of the state-feedback control for an 8-PUS parallel robot. The two examples present the applications of the new algorithm and demonstrate its validity and efficiency.

Constrained Autonomous Precision Landing via Dual Quaternions and Model Predictive Control

The problem of powered descent guidance and control for autonomous precision landing for next-generation planetary missions is addressed. The precision landing algorithm aims to trace a fuel-optimal trajectory while keeping geometrical constraints such as the line of sight to the target site. The design of an autonomous control algorithm managing such mission scenarios is challenging due to fact that critical geometrical constraints are coupled with the translational and rotational motions of the lander spacecraft, leading to a complex motion-planning problem. This problem is approached within the model predictive control framework by representing the dynamics of the rigid body in a uniform gravity field via a piecewise affine system taking advantage of the unit dual-quaternion parameterization. Such a parameterization in turn enables a six-degree-of-freedom motion planning in a unified framework while also admitting a quadratic cost on the required control commands to minimize propellant consumption. A novel feature of the proposed approach is the development of convex representable subsets in the configuration space in terms of unit dual quaternions. The stability and reachability issues of the corresponding piecewise affine model predictive control approach are then discussed. Numerical simulations are presented to demonstrate the effectiveness of the proposed methodology for autonomous precision landing.

The cross-motion invariant group and its application to kinematics

This article presents the cross-motion invariant group—CMI(3)—whose group operation is defined over unit dual quaternions such that rigid motions are cross-motion invariant; that is, the resultant translation does not depend on rotation and vice-versa. We present the main properties of CMI(3) and the differences between this group and the standard group Spin(3) R 3 of unit dual quaternions, as well as the kinematic equations under a sequence of CMI(3) operations. Two numerical examples are presented in order to illustrate the main characteristics of CMI(3).