# Euler-Angle-Joints (EAJs)

01 Jan 2013-pp 27-55

TL;DR: In this article, the Euler and Bryant angles are used to represent a sequence of rotations about the same axis separated with a rotation about a different axis, denoted as α-β-γ.

Abstract: As frequently noted in the literature on robotics (Sugihara et al. 2002; Kurazume et al. 2003; Vukobratovic et al. 2007; Kwon and Park 2009) and mechanisms (Duffy 1978; Chaudhary and Saha 2007), a higher Degrees-of-Freedom (DOF) joint, say, a universal, a cylindrical or a spherical joint, can be represented using a combination of several intersecting 1-DOF joints. For example, a universal joint also known as Hooke’s joint is a combination of two revolute joints, the axes of which intersect at a point, whereas a cylindrical joint is a combination of a revolute joint and a prismatic joint. Similarly, the kinematic behavior of a spherical joint may be simulated by the combination of three revolute joints whose axes intersect at a point. The joint axes can be represented using the popular Denavit and Hartenberg (DH) parameters (Denavit and Hartenberg 1955). For the spherical joints, an alternative approach using the Euler angles can also be adopted, as there are three variables. For spatial rotations, one may also use other minimal set representations like Bryant (or Cardan) angles, Rodriguez parameters, etc. or non-minimal set representation like Euler parameters, quaternion, etc. The non-minimal sets are not considered here due the fact that the dynamic models obtained in this book are desired in minimal sets. The minimal sets, other than Euler/Bryant angles, are discarded here as they do not have direct correlation with the axis-wise rotations. It is worth mentioning that the fundamental difference between the Euler and Bryant angles lies in a fact that the former represents a sequence of rotations about the same axis separated with a rotation about a different axis, denoted as α–β–α, whereas the latter represents the sequence of rotations about three different axes, denoted as α–β–γ. They are also commonly referred to as symmetric and asymmetric sets of Euler angles in the literature. For convenience, the name Euler angles will be referred to both Euler and Bryant angles, hereafter.

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01 Jan 2014

TL;DR: Three-dimensional kinematics of the glenohumeral (GH) joint and thorax following supraspinatus repair support the use of thorax motion to compensate for limited GH joint mobility, however even with compensatory motion RC repair subjects completed tasks with similar temporal quality as those without shoulder pathology.

Abstract: KINEMATIC ANALYSIS OF THE GLENOHUMERAL JOINT: A COMPARISON OF POST-OPERATIVE ROTATOR CUFF REPAIR PATIENTS AND CONTROLS Ryan R. Inawat, B.S Marquette University, May 2014 Rotator cuff (RC) repair is a standard surgical intervention used to alleviate pain and loss of function in the shoulder due to torn RC tendons, involving re-attachment of the tendon to the humerus. Quantitative evaluation of kinematics following RC repair is possible with video motion analysis techniques, yet is rarely performed. With the purpose of quantifying the effects of RC repair, a Vicon 524 (Oxford, UK) motion analysis system was used to investigate three-dimensional (3D) kinematics of the glenohumeral (GH) joint and thorax following supraspinatus repair. A validated, 18 marker, inverse dynamics model based on ISB standards was applied to analyze GH joint kinematics in a population of persons who underwent recent RC repair and persons with ideal shoulder health. The kinematic data characterized GH joint motion during ADLs following single tendon repair of the supraspinatus. Motion capture was performed on ten (10) healthy subjects and ten (10) subjects at 9 to 12 weeks post arthroscopic RC tendon repair (supraspinatus). The tasks included ten ADLs characteristic of motions normally performed at home and work and three rehabilitation motions performed both actively and passively. Kinematics of the GH joint and thorax, as well as temporal characteristics of the trials were analyzed between groups. Hotelling’s T test and Welch’s t-test were used to examine significant differences in triplanar (3D) kinematics between the groups (α = 0.05). ADLs with significantly different kinematics suggest that specific combined motions (e.g. performing extension while adducting as done when reaching to perineum) may be limited after rotator cuff repairs (especially after repairs of the supraspinatus), while single-plane mobility is returned to a healthy range suitable for most ADLs. Significantly different thorax kinematics support the use of thorax motion to compensate for limited GH joint mobility, however even with compensatory motion RC repair subjects completed tasks with similar temporal quality as those without shoulder pathology.

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##### References

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01 Jan 1986

TL;DR: This chapter discusses Jacobians: Velocities and Static Forces, Robot Programming Languages and Systems, and Manipulator Dynamics, which focuses on the role of Jacobians in the control of Manipulators.

Abstract: 1. Introduction. 2. Spatial Descriptions and Transformations. 3. Manipulator Kinematics. 4. Inverse Manipulator Kinematics. 5. Jacobians: Velocities and Static Forces. 6. Manipulator Dynamics. 7. Trajectory Generation. 8. Manipulator Mechanism Design. 9. Linear Control of Manipulators. 10. Nonlinear Control of Manipulators. 11. Force Control of Manipulators. 12. Robot Programming Languages and Systems. 13. Off-Line Programming Systems.

5,992 citations

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3,862 citations

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TL;DR: Two computationally efficient algorithms are presented for determining three-axis attitude from two or more vector observations that are useful to the mission analyst or spacecraft engineer for the evaluation of launch-window constraints or of attitude accuracies for different attitude sensor configurations.

Abstract: Two computationally efficient algorithms are presented for determining three-axis attitude from two or more vector observations. The first of these, the TRIAD algorithm, provides a deterministic (i.e., nonoptimal) solution for the attitude based on two vector observations. The second, the QUEST algorithm, is an optimal algorithm which determines the attitude that achieves the best weighted overlap of an arbitrary number of reference and observation vectors. Analytical expressions are given for the covariance matrices for the two algorithms using a fairly realistic model for the measurement errors. The mathematical relationship of the two algorithms and their relative merits are discussed and numerical examples are given. The advantage of computing the covariance matrix in the body frame rather than in the inertial frame (e.g., in terms of Euler angles) is emphasized. These results are valuable when a single-frame attitude must be computed frequently. They will also be useful to the mission analyst or spacecraft engineer for the evaluation of launch-window constraints or of attitude accuracies for different attitude sensor configurations.

1,394 citations

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07 Aug 2002

TL;DR: A real-time motion generation method that controls the center of gravity (COG) by indirect manipulation of the zero moment point (ZMP) and provides humanoid robots with high-mobility.

Abstract: A humanoid robot is expected to be a rational form of machine to act in the real human environment and support people through interaction with them. Current humanoid robots, however, lack in adaptability, agility, or high-mobility enough to meet the expectations. In order to enhance high-mobility, the humanoid motion should be generated in real-time in accordance with the dynamics, which commonly requires a large amount of computation and has not been implemented so far. We have developed a real-time motion generation method that controls the center of gravity (COG) by indirect manipulation of the zero moment point (ZMP). The real-time response of the method provides humanoid robots with high-mobility. In the paper, the algorithm is presented. It consists of four parts, namely, the referential ZMP planning, the ZMP manipulation, the COG velocity decomposition to joint angles, and local control of joint angles. An advantage of the algorithm lies in its applicability to humanoids with a lot of degrees of freedom. The effectiveness of the proposed method is verified by computer simulations.

508 citations

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07 Apr 1986

TL;DR: A new geometric notation for the description of the kinematic of open-loop, tree and closed-loop structure robots is presented, derived from the well-known Denavit and Hartenberg (D-H) notation.

Abstract: This paper presents a new geometric notation for the description of the kinematic of open-loop, tree and closed-loop structure robots. The method is derived from the well-known Denavit and Hartenberg (D-H) notation, which is powerful for serial robots but leads to ambiguities in the case of tree and closed-loop structure robots. The given method has all the advantages of D-H notation in the case of open-loop robots.

400 citations