Modelling Cable-driven Continuum Robots and their applications

Abstract

Flexible robots have been gaining traction in real-world applications over their rigid counterparts for their favourable features such as flexibility, compliance, better manoeuvrability, lightweight construction and ability to work in constricted environments in the presence of obstacles. Continuum robots are currently one of the most used flexible robots. These robots are characterized by a flexible backbone that is actuated by various means, of which the ones actuated by cables are widely used for their simplicity in design, ease of construction and operation. These robots, called `Cable-driven Continuum Robots’ (CCR), consist of a flexible backbone to which multiple disks are attached. Cables are passed through holes in the disk sequentially from the base to the tip, and when pulled, the flexible backbone and the CCR can attain different shapes based on their cable routing and backbone configuration. This thesis focuses on the kinematics and statics modelling of CCRs with different configurations and presents some of their applications.

First, the forward kinematics of the CCR is explored. Due to its geometry, a CCR with a straight routed cable can be discretized into sections, and in one approach in this work, we model each section as a four-bar mechanism. Using the geometry of the actuated CCR, the pose of the CCR was determined by minimization of the coupler angles of this imaginary four-bar. This was then extended to multiple straight and generally routed cables by assuming a second imaginary four-bar, enabling us to track motion in 3D. A method based on the minimization of strain energy is then proposed, which is shown to be equivalent to the minimization of coupler angles. The strain energy-based method is extended to generally routed cables by appropriately adding terms due to torsion. The strain energy-based formulation is shown to be applicable to CCR with a pre-bent backbone (without any pre-tension). The proposed optimization-based method is geometry-based, requiring only the Poisson’s ratio and the cable actuation length as the input parameter. The proposed four-bar and the strain energy-based methods are verified experimentally using multiple 3D-printed CCRs, and there is good agreement between the simulations and experiments. The method is also compared to another well-known approach using the Cosserat rod model and is shown to provide equivalent accuracy in prediction while taking lower simulation times in most cases.

The optimization-based approach is easily amenable to obtaining the behaviour of a CCR when there are obstacles present in the workspace. The obstacles are modelled with inequality constraints, and the locations of contact of the CCR with the obstacles are obtained by observing the Lagrange multiplier associated with the inequality constraints -- when in contact, the Lagrange multiplier becomes much larger. A method is devised to predict the static force exerted by the obstacle on the CCR based on the principle of virtual work. The results obtained from the modified optimization method, Lagrange multipliers and the principle of virtual work are verified in experiments using a load cell.

The optimization-based forward kinematics is used to solve two different inverse problems, namely, to achieve a desired final shape of the backbone, what could be the possible general cable routing, and what are the possible initial shapes of the backbone. The latter algorithm is used to design a three-fingered gripper to grasp a desired object, ensuring optimal grasp and multiple points of contact.

The aforementioned forward kinematics model is used to train multiple neural networks to solve the inverse kinematics of CCR with varying cable routings and geometry for a single straight-routed cable with both fixed and varying geometry. The inverse problem for general cable routing - finding the cable routing to achieve a desired end effector position is also explored by using a trained forward kinematics model of the CCR and a reconstruction-based loss function, tackling the issue of multiple solutions. Later, a comparison is provided between the optimization-based and neural network approaches for the inverse problems.