Continuum robotic manipulators articulate due to their inherent compliance. Tendon actuation leads to compression of the manipulator, extension of the actuators, and is limited by the practical constraint that tendons cannot support compression. In light of these observations, we present a new linear model for transforming desired beam configuration to tendon displacements and vice versa. We begin from first principles in solid mechanics by analyzing the effects of geometrically nonlinear tendon loads. These loads act both distally at the termination point and proximally along the conduit contact interface. The resulting model simplifies to a linear system including only the bending and axial modes of the manipulator as well as the actuator compliance. The model is then manipulated to form a concise mapping from beam configuration-space parameters to n redundant tendon displacements via the internal loads and strains experienced by the system. We demonstrate the utility of this model by implementing an optimal feasible controller. The controller regulates axial strain to a constant value while guaranteeing positive tendon forces and minimizing their magnitudes over a range of articulations. The mechanics-based model from this study provides insight as well as performance gains for this increasingly ubiquitous class of manipulators.

Mechanics Modeling of Tendon-Driven Continuum Manipulators

This paper deals with the wrench-feasible workspace (WFW) of n-degree-of-freedom parallel robots driven by n or more than n cables. The WFW is the set of mobile platform poses for which the cables can balance any wrench of a given set of wrenches, such that the tension in each cable remains within a prescribed range. Requirements of nonnegative cable tensions, as well as maximum admissible tensions, are thus satisfied. The determination of the WFW is an important issue since its size and shape are highly dependent on the geometry of the robot and on the ranges of allowed cable tensions. The approach proposed in this paper is mainly based on interval analysis. Two sufficient conditions are presented, namely, a sufficient condition for a box of poses to be fully inside the WFW and a sufficient condition for a box of poses to be fully outside the WFW. These sufficient conditions are relevant since they can be tested, with the means to test them being discussed in the paper. Used within usual branch-and-prune algorithms, these tests enable WFW determinations in which full-dimensional sets of poses (volumes) are found to lie within or, on the contrary, to lie outside the WFW. This provides a useful alternative to a basic discretization, the latter consisting of testing a discrete (zero-dimensional) finite set of poses. In order to improve the efficiency of the computations, a means to mitigate the undesirable effects of the so-called wrapping effect is introduced. The paper also illustrates how the proposed approach is capable of dealing with small uncertainties on the geometric design parameters of a parallel cable-driven robot.

https://hal-lirmm.ccsd.cnrs.fr/lirmm-00573491/document

Interval-Analysis-Based Determination of the Wrench-Feasible Workspace of Parallel Cable-Driven Robots

Tendon-based Stewart platforms are a concept for innovative manipulators where the load to move almost coincides with the payload. After an overview over the state of research and some concepts of kinematics (singularity and redundancy), the thesis discusses aspects of the technically usable workspace (positive tendon forces, limits of tension, singularity, stiffness, collisions between tendens). A representation of the controllablwe workspace by means of polynomial inequalities is developed. 
Optimal solutions are provided to the problem of finding appropriate force distributions in the tendons. These solutions can be discontinuous in time, but they can be approximated with continuous ones. An algorithm is given for this. 
From these results, a quality measure for workspace is derived and used to state design rules which help achieving good workspaces. For some systems, sample trajectories are simulated.

Analysis of the Workspace of Tendon-based Stewart Platforms

Redundant actuation of parallel manipulators can lead to internal forces without generating end-effector forces (preload). Preload can be controlled in order to prevent backlash during the manipulator motion. Such control is based on the inverse dynamics. The general solution of the inverse dynamics of redundantly actuated parallel manipulators is given. For the special case of simple overactuation an explicit solution is derived in terms of a single preload parameter. With this formulation a computational efficient open-loop preload control is developed and applied to the elimination of backlash. Its simplicity makes it applicable in real-time applications. Results are given for a planar 4RRR manipulator and a spatial heptapod.

Internal Preload Control of Redundantly Actuated Parallel Manipulators&#8212;Its Application to Backlash Avoiding Control

Robust PID control of fully-constrained cable driven parallel robots

This paper presents the motion control of a six degree-of-freedom tendon-based parallel manipulator, which moves a platform with high speed using seven cables. To control the motion of the platform along desired trajectories in space, nonlinear feedforward control laws in the cable length coordinates are used. Taking account of the effect of redundancy on actuation, the optimal tension distribution should be considered to the advantage of the control laws. Using a method based on the analysis of the workspace condition, tension constraints and limiting torque constraints of actuators, an analytical solution for optimum tension distribution was found and used to compute the force in each cable for compensation of dynamic errors. It is experimentally demonstrated that the proposed control laws reduce the energy consumption of the actuators and satisfy the path tracking accuracy.

Motion control of a tendon-based parallel manipulator using optimal tension distribution

Design, analysis and realization of tendon-based parallel manipulators.

Tendon-based Stewart platforms are capable of high speeds and accelerations.
Thus, a reliable controller running with a high frequency is required. Here, an
implementation based on a modular controller architecture is shown. To control the platform on a given trajectory, the implemented position control has to be extended by a tendon force control which generates defined and safe force values. In the case of one redundant tendon, a direct calculation is possible. Having at least two redundant tendons, the computation of a force distribution requires optimization methods, which are numerically expensive. In this paper, algorithms for both cases and their embedding into the controller are presented.

A Modular Controller for Redundantly Actuated Tendon-Based Stewart Platforms

Compared to serial robots, parallel kinematic machines (PKM) offer superior properties. However, modelling and analysis of PKMs are challenging tasks due to their complicated kinematic structure. In this paper methods are presented to analyze the workspace of a class of 6 degree-of-freedom (dof) PKMs with constant leg length. These methods apply interval analysis in order to calculate the orientation workspace and to verify if a certain machine covers a desired workspace. The machines are assembled from a construction kit that defines components like legs, frames, and platforms. In this paper algorithms are presented to calculate and to validate the workspace of parallel kinematic machines with constant leg length. Due to the application of interval analysis the validity of the results is guaranteed. Equations have been introduced that take into account the link interference and the limits on both articulated and passive joints. The algorithms supply orientation workspace calculation and verification at a short runtime. Therefore, it is capable for the optimization of the parameters of the Linapod which requires a large number of function evaluations

Orientation workspace verification for parallel kinematic machines with constant leg length. Verifikation eines Orientierungsarbeitsplatzes für parallelkinematische Maschinen mit konstanter Beinlänge

This paper presents a method to numerically linearise a complex mechanicalsystem. The method is applied to the parameter optimisation of a controlled elastic hydro-mechanical system. The virtual spring damper control concept is used to damp the oscillating system. The system dynamics of the full system are modelled as an interdisciplinary model. From this a mechanical substitution model is derived and linearised at a given operating point. The system matrix of the linearised system is used to efficiently calculate an optimisation criterion that leads to optimal control parameters for rapid oscillation damping. To verify the results a classical, but very time consuming, optimisation is performed. The quality of the optimal parameter sets found is evaluated by comparing simulation results of the complex interdisciplinary model. The system response is almost the same in both cases. The parameters found using the linear model are only slightly different to the parameters found using the non-linear and interdisciplinary model. The performance of the linear model based optimisation is good – it reduces the computational cost by a factor of more than one hundred.

Daniel Franitza

Papers

Motion control of a tendon-based parallel manipulator using optimal tension distribution

Design, analysis and realization of tendon-based parallel manipulators.

A Modular Controller for Redundantly Actuated Tendon-Based Stewart Platforms

Orientation workspace verification for parallel kinematic machines with constant leg length. Verifikation eines Orientierungsarbeitsplatzes für parallelkinematische Maschinen mit konstanter Beinlänge

Numerical Linearisation Method to Efficiently Optimise the Oscillation Damping of an Interdisciplinary System Model