Kinematic modeling and control of a robot arm using unit dual quaternions

https://doi.org/10.1016/j.robot.2015.12.005Get rights and content

Highlights

  • Kinematic modeling and pose control of multi-dof robotic arms.

  • Compact and simple formulation.

  • Use of unit dual quaternions and its algebra.

Abstract

This paper exploits screw theory expressed via unit dual quaternion representation and its algebra to formulate both the forward (position+velocity) kinematics and pose control of an n-dof robot arm in an efficient way. Efficiency is in less computer memory usage, in fast computation of the equations, in singularity-free representation of task space, in robustness to numerical errors, and in compactness of the representations. The formulation is simple, intuitive and straightforward to implement. We validated this formulation experimentally on a 7 dof robot arm.

Introduction

Unit dual quaternion (UDQ) representation of a pose (position + orientation) has been receiving a lot of attention from the robotics community both for kinematic modeling and control purposes  [1], [2], [3], [4], [5], [6], [7], [8] recently, although its storage and computational efficiencies over the homogeneous transformation matrix (HTM) have been known since more than two decades  [9], [10]. The study in  [11] shows the superior performance of UDQ over HTM in kinematic modeling of an n-dof robot arm, and recently in  [12] for the proportional control design. Other appealing advantages of UDQ are singularity free representation of the Euclidean space, robustness to numerical errors and compactness of the representation. UDQ has also been effectively used in computer graphics  [13], in computer-aided design  [14], in computer vision  [15], in navigation  [16] and so on.

The most well-known method for robot kinematics is based on the Denavit and Hartenberg (DH) notation  [17] and the homogeneous transformation of the points through HTM  [18]. So far all the existing works  [4], [5], [6], [11] on modeling of robot kinematics with UDQ continue to follow DH approach. We think that DH wastes some capacity of UDQ, since DH’s first design is based on point transformations with HTM.

In this paper, for kinematic modeling, we followed the screw theory approach based on line transformations presented in  [19], and we adapted it to unit dual quaternion representation and its algebra, since UDQ has been found as the most compact and efficient way of expressing screw displacement  [9], [10]. For kinematic control purposes, we used the logarithm of an error unit dual quaternion as a generalized proportional control law which is first introduced in  [1] and we also analyzed its global stability in terms of the screw parameters’ value ranges. Definition of the pose error between the two pose unit dual quaternions should be done through the multiplication operator of the dual quaternion algebra rather than the subtraction operator as it is done in  [5], [6] which is not correct (although the stability of the control law is proven). Some recent works  [7], [8] exploited UDQ for the design of robust control laws and for the flexible modeling of the cooperative task spaces by passing to 8 manifold to obtain the lacking commutative property back through Hamilton operators (8×8 matrices), however leaving the computational advantages of UDQ algebra away. One might also think to use Rodrigues’ efficient rotation formula through a rigid-body pose represented by a 3D translation vector and a 4D rotation vector with Rodrigues’ axis–angle parameters. We here name this representation as TAA. We remark that TAA has a singularity. Whenever the resulting angle in TAA is zero, the axis part of the rotation representation is undetermined  [20]. Table 1 lists the storage and computational cost requirements for a rigid-body transformation under 4 different representations: homogeneous transformation matrices (HTM), unit dual quaternions with Hamilton operators (UDQwH), pose with Rodrigues’ parameters (TAA) and unit dual quaternions (UDQ). Although TAA needs lesser storage, we note that it requires 7 trigonometric functions and 1 square-root function computations more on top of what is listed in Table 1. TAA also lacks an efficient algebra.

This paper combines efficiently all the advantages of screw theory based on UDQ and its algebra for kinematic modeling and pose control of a robot arm and experimentally validates it. Each relevant work, in the reviewed literature, somehow misses one point in combining all these together as it is discussed above. We then list the contributions of this paper as follows:

  • All the advantages (i.e., compactness, storage, computational efficiency, etc.) of unit dual quaternion representation and its algebra are exploited.

  • The forward position kinematics (FPK), for the first time, is written in the dual space with the product of exponentials (POE) formula of screw theory by replacing the matrix exponentials with the unit dual quaternions. Everything is expressed in one single reference frame (i.e., robot home frame). This makes FPK simpler and more intuitive. Consequence of this formulation is that the computation of the robot Jacobian is straightforward and fast.

  • The kinematic modeling and the pose control problems of a robot arm are solved compactly with fewer number of arithmetic operations and storage requirements than many of the existing relevant approaches proposed in the robotics literature.

  • Correctness of the proposed kinematic modeling and control approaches is validated experimentally on a 7 dof robot arm.

  • All the variables and equations are explained clearly and without any ambiguity. That is to say, for example, a pose variable is precisely stated with in which frame it is defined and with respect to which frame it is expressed. The paper is also self-sufficient such that one can implement everything presented here without looking for any other relevant reference or book.

The rest of the paper goes on as follows: Section  2 explains the pose (position + orientation) representation of the end-effector, the forward position and velocity kinematics of the robot; Section  3 first defines the pose error, later it proposes a control law to regulate this pose error, and finally it analyzes the stability of the proposed control law; Section  4 validates experimentally the proposed kinematic modeling and control theory on the 7 dof Kuka robot arm; finally Section  5 concludes the paper.

We also note that, for better understanding of the paper, the reader can look up to Appendix for further information about the quaternions, dual numbers, and dual quaternions.

Section snippets

Pose representation

We represent the position and orientation of the end-effector of a robot arm with a unit dual quaternion  [13], [15], [21]: xˆ=exp(θˆ2sˆ)=cos(θˆ2)+sˆsin(θˆ2) where θˆD and sˆD3×1 are respectively the dual angle and the unit dual vector of a directed 3D line: θˆ=θ+εd,sˆ=+εm,ε2=0,ε0. Above, {θ,d,,m} are the screw displacement parameters. θ is a rotation angle around the screw axis, d is a translation along the same screw axis, is the unit direction vector of this screw axis, and m is the

End-effector pose error

We define the error unit dual quaternion, eˆ, as the difference between the current end-effector pose at a and the desired end-effector pose at ad in the home frame a0: eˆ=a0xˆa0aa0xˆa0ad where a0xˆa0a is the current end-effector pose and a0xˆa0ad is the inverse of the desired end-effector pose a0xˆa0ad which is obtained through the classical quaternion conjugate of a dual quaternion.

Control law

We define the Cartesian control law a0ξˆa0a in the dual space in terms of the logarithm of the error unit dual

Experiments

The presented formulation is validated on a Kuka LWR IV seven dof robot arm which is equipped with a Shadow dexterous hand  [22]. In the experiment, we first reach to grasp a bottle lying on a table from a known pose, then after grasping we correct the posture of the bottle and put it back. In Fig. 2 left image shows the initial configuration the Kuka robot arm plus the Shadow dexterous hand and the bottle lying on the table. In Fig. 2 middle image shows the desired reach pose of the robot arm,

Conclusions

This paper used unit dual quaternions to model the kinematics and then to control the pose of a robot arm. Modeling is compact and fast. Therefore, computation of the control law is fast. Besides, the task space is singularity free. This formulation provides an important advantage if one uses it to model and control a robotic system which has many degrees of freedom, such as a humanoid robot.

This work may provide a basis for future research on dynamic modeling and control of robot arms in a

Erol Özgür received the Ph.D. degree in Robotics and Vision from the University of Blaise Pascal, France, in 2012. Between 2012 and 2014, he was a postdoctoral fellow in Pascal Institute—UBP/CNRS/IFMA, France. Since 2015, he is an assistant professor in Université d’Auvergne. His research interests are vision-based robot control and computer vision.

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    Erol Özgür received the Ph.D. degree in Robotics and Vision from the University of Blaise Pascal, France, in 2012. Between 2012 and 2014, he was a postdoctoral fellow in Pascal Institute—UBP/CNRS/IFMA, France. Since 2015, he is an assistant professor in Université d’Auvergne. His research interests are vision-based robot control and computer vision.

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