The study of the power of a point with respect to a circle and its application to orthogonal circles, bundles of circles, etc., has received a lot of attention in the past. In this paper, we show how the concept of conjugate ellipses generalizes the concept of orthogonal circles. It is also shown that it is possible to design 3R serial regional robots whose inverse kinematics can be reduced to the computation of the intersection between two conjugate ellipses which, in turn, can be reduced to the intersection of an ellipse and a line by relying on the concept of radical conic. The relevance of these findings is illustrated through an example.

The intersection of circles appears as a frequent subproblem in the domains of computational kinematics and geometry. For this reason, the methods for computing its solutions need to be stable and simple. This paper surveys the solution method for circle intersection given by the approach of distance geometry via Cayley–Menger bideterminants. In particular, the equivalence of the squared-quantity method to its linear-quantity counterpart is shown and novel interconnections to related concepts of Non-Euclidean geometry are worked out.

Due to high dimensions and nonlinear behavior, the characterization of the entire workspace of a spatial robotic system represents an ambitious computation task.

The characterization of the workspace for general spatial 3R chains with skew joint axes is refined by describing the variety of singular displacements as the union of the singular configuration manifolds of orientation type, position type, and attitude type. The surface of attitude singularities is revealed by transferring the singularity sets of spherical 3R chains with intersecting joint axes to the geometry of spatial kinematic chains with skew, non-intersecting joint axes. For this purpose, the degeneracy of a screw system is analyzed by means of a particular angle concept for a set of three oriented lines in space. The obtained argumentation is expressed in terms of geometric manipulator Jacobians completing previous results.

This paper introduces methods to determine the exponential coordinates of first kind and of second kind for an arbitrary spatial displacement given in terms of the left adjoint representation. Due to the algebraic properties of the \((6 \times 6)\)-matrix group, the obtained formulae are structure-preserving generalizations of their purely-rotative counterparts in coherence with the principle of transference. While the exponential coordinates of first kind represent the line-geometric parameters of a displacement (rotation and translation along a unit spear), the exponential coordinates of second kind coincide with dual Euler angles and with the parameters according to the kinematic convention by Sheth and Uicker. In either case, a spatial displacement is specified via six independent scalars, in form of a dual angle and a dual unit vector (\(2+4=6\)) for the first kind and in form of three dual angles (\(3\cdot 2 =6\)) along three sequentially orthogonal axes for the second kind. From a practical viewpoint, the parametrization method enables an automated parametrization of the kinematics of an arbitrary mechanism. From a theoretical viewpoint, the reported methods are relevant due to their structural simplicity and coherence.

Based on the representation of rigid body displacements as adjoint matrices, the article introduces the adjoint trigonometric representation of displacements (ATRD) as a further generalization of the trigonometric representation of rotations. In comparison to the dual Rodrigues–Euler–Gauß–Gelman equation, recently reported for affine screw displacements with arbitrary, fixed pitches, the ATRD is built upon a product of a unit line and a dual angle, instead of upon a product of a unit screw and a real angle. Due to this conceptual difference, the ATRD requires four independent parameters of a unit line instead of five when parametrizing a displacement along a unit screw. As a consequence for computational kinematics, the ATRD permits transferring the analytic solution to the inverse kinematics problem (IKP) of 3‐DOF, general, spherical 3R‐chains into a closed‐form solution to the IKP of 6‐DOF, general, affine 3C‐chains.

The novel mechanism Active Ankle features three degrees of freedom that operate in an almost spherical manner. In comparison to spherical devices, its design offers advantages such as high stiffness, a simple and robust construction, and a good stress distribution. In the present paper, a comprehensive study of the design, analysis, and control of the Active Ankle in its almost-spherical work modality is provided. In particular, the kinematic analysis of the mechanism is conducted, solving the full inverse, the rotative inverse, and the forward kinematic problems. In addition, the manipulator's workspace is characterized and the kinematic control, that has been implemented on a prototype of Active Ankle, is presented together with experimental results that demonstrate the employability as an ankle joint in a full body exoskeleton.

This paper formally documents a method that enables the scaling of displacements, elements of the group of rigid body motions, in all their six degrees of freedom individually in linear manner. The method incorporates in particular a suitable modulation method for the rotational part of a displacement. This simple, yet non-trivial function has not been reported in the literature as of today. The rotation scaling relies on the representation of a rotation in terms of its angle and its axis as given via Euler’s rotation theorem. It is argued that the overall workspace scaling falls in the class of ‘regularized linear functions’ in the tangent space.

Active Ankle is a novel 3 DoF parallel mechanism which works in an almost spherical manner. Its geometry provides various advantages like good stress distribution, low link diversity and robust construction. Determining all the solutions to the direct kinematics problem is an important and challenging step in kinematic analysis of any newly invented parallel manipulator due to the coupled nature of the constraint equations. In this paper, we make use of powerful methods in computational algebraic geometry to provide a rational univariate representation of direct kinematics solution in the form of a \(40^\circ \) univariate polynomial. In the presented analysis, up to 16 real solutions of the direct kinematics problem for this mechanism have been obtained. In addition, the results of its torsional motion analysis are presented and singularities of the mechanism are highlighted during this motion. Also, the assembly modes where this mechanism behaves as an almost-spherical device are identified, which is the main contribution of the paper.

In this paper it is demonstrated that the solution space of the inverse kinematic problem of an anthropomorphic, redundant 7R chain for a given pose does consist of eight different coinciding circles instead of a single circle that has been reported as of today. By modeling the structure using the convention by Sheth and Uicker, the displacements within the kinematics of the chain are partitioned in time-invariant displacements along rigid links and time-variant displacements along the seven rotative joints. In particular, the subchains of shoulder, elbow, and wrist are preserved. By respecting the 'flips' of these three substructures the eight-fold occupancy of the redundancy circle is obtained. The result corresponds to the eight IK solutions for regional-spherical arms and provides a prerequisite for using all capabilities of respective robots in practical applications.

Based on new concepts for three (6 × 6) -matrix operators for Plücker vectors, the paper presents a novel adjoint formulation of the Rodrigues equation for spatial displacements in the trigonometric form. Due to the structural similarity of this dualized equation to its spherical pendant, the principle of transference is applicable to lift the analytic solution to the inverse kinematics problem (IKP) of generic spherical 3R chains yielding a simple analytic solution for generic spatial 3R chains based on the geometry of lines and screws.

In robotic space missions, different challenges like high latency, unknown environments or bad visual conditions make it hard for the operators to plan and execute the mission. We present our mission control using virtual reality to remotely execute a sample-return mission in the field via satellite.

The design of various robots in industrial and academic contexts integrates closed loops to improve the mechanical stiffness in comparison with purely serial or tree-type topologies. In particular, planar kinematic loops as parallelograms or double parallelograms are employed in such hybrid robots. Since these systems are geometrically overconstrained in the group of spatial Euclidean motions, the computational performance and numerical accuracy of any model-based dynamics software is negatively affected. This paper introduces a novel method to avoid these numerical issues for any hybrid system with loops that can be characterized by the concept of linear mimic joints: these are passive joints which depend on an active joint in a closed loop in a linear manner. With the proposed approach, the loop closure functions are automatically composed from the robot description file and integrated into the analytical equations for solving the forward and the inverse dynamics problems. The paper illustrates the application of this method for a novel shoulder mechanism containing a planar six bar mechanism that has been designed for the Recupera whole-body exoskeleton.

Since parallel manipulators only provide restricted workspaces in comparison to their serial counterparts they cannot compete as versatile multi-purpose tools in flexible industrial setups. However, their superior properties in terms of stiffness, payload, speed, and acceleration often allow an advantageous application in more tailored use-cases. This paper lays out the concept of a novel parallel mechanism - called the ACTIVE ANKLE - which operates in an almost-spherical manner with three degrees of freedom. The text motivates the primarily intended application of the novel device as an ankle joint of a full-body exoskeleton. In addition, the paper discusses the design, the topology, results of a motion simulation, and a comparison with related mechanisms of the ACTIVE ANKLE.

In this paper, the concept of the innovative exoskeleton for stroke rehabilitation of the Recupera-Reha project is presented. By applying innovative electromechanical solutions, advanced control approaches, and by using biosignal data, the system will be well-fitted. Furthermore, it will behave transparently to the user and will support different training modalities. The therapy situation will be supported and exercises in the field of everyday practice in the private or professional environment will be enabled. As an autonomous treatment system, it will give assistance as much as needed and will enable the patient to make therapies across sector borders (rehab-outpatient-work) for the first time.

For the computation of rigid body dynamics, the Newton-Euler equations represent a crucial relation unifying the laws of motion by Newton and Euler using the language of instantaneous screws.Typically, Newton-Euler equations are stated in spatial or in body coordinates, respectively.In this paper, a general formulation of Newton-Euler equations is provided for arbitrary reference systems.In particular, the general form unifies the known equations in spatial and body coordinates.To the best of the authors' knowledge, this relation between the spatial and the body form has not been reported in literature.The novel formulation is based on the concept of time differentiation with respect to moving reference systems.

In this paper, a novel view on complex numbers and quaternions is presented by introducing a five-dimensional complex space which is defined as the 'union' of the complex plane C and the quaternion space H. It is demonstrated how the complex 5-space can be visualized by R-3 and by R-2 for rotations with a fixed rotation axis. In these visualizations, the algebraic representations of a rotation, using a complex number, quaternions, and a rotation matrix, appear in an elementary-geometric setup which is generalizing the unit circle. The definition of the complex 5-space is based on an explicit distinction of four different imaginary units. The embedding of a rotation matrix into the three-dimensional view is achieved by the choice of an appropriate basis for the representing matrix of the rotation.